{"id":3587,"date":"2024-06-28T17:10:31","date_gmt":"2024-06-28T17:10:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3587"},"modified":"2024-08-04T11:07:59","modified_gmt":"2024-08-04T11:07:59","slug":"more-basic-functions-and-graphs-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/more-basic-functions-and-graphs-cheat-sheet\/","title":{"raw":"More Basic Functions and Graphs: Cheat Sheet","rendered":"More Basic Functions and Graphs: Cheat Sheet"},"content":{"raw":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+More+Basic+Functions+and+Graphs.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\r\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\r\n<h2>Essential Concepts<\/h2>\r\n<p><strong>Trigonometric Functions<\/strong><\/p>\r\n<ul id=\"fs-id1170572167654\">\r\n\t<li>Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180\u00b0 has a radian measure of [latex]\\pi [\/latex] rad.<\/li>\r\n\t<li>For acute angles [latex]\\theta[\/latex], the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is [latex]\\theta[\/latex].<\/li>\r\n\t<li>For a general angle [latex]\\theta[\/latex], let [latex](x,y)[\/latex] be a point on a circle of radius [latex]r[\/latex] corresponding to this angle [latex]\\theta[\/latex]. The trigonometric functions can be written as ratios involving [latex]x, \\, y[\/latex], and [latex]r[\/latex].<\/li>\r\n\t<li>The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period [latex]2\\pi[\/latex]. The tangent and cotangent functions have period [latex]\\pi[\/latex].<\/li>\r\n<\/ul>\r\n<p><strong>Inverse Functions<\/strong><\/p>\r\n<ul id=\"fs-id1170572470433\">\r\n\t<li>For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.<\/li>\r\n\t<li>If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.<\/li>\r\n\t<li>For a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f^{-1}[\/latex] and [latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\r\n\t<li>Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.<\/li>\r\n\t<li>The graph of a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] are symmetric about the line [latex]y=x[\/latex].<\/li>\r\n<\/ul>\r\n<p><strong>Exponential and Logarithmic Functions<\/strong><\/p>\r\n<ul id=\"fs-id1170572216270\">\r\n\t<li>The exponential function [latex]y=b^x[\/latex] is increasing if [latex]b&gt;1[\/latex] and decreasing if [latex]0&lt;b&lt;1[\/latex]. Its domain is [latex](\u2212\\infty ,\\infty)[\/latex] and its range is [latex](0,\\infty)[\/latex].<\/li>\r\n\t<li>The logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of [latex]y=b^x[\/latex]. Its domain is [latex](0,\\infty)[\/latex] and its range is [latex](\u2212\\infty,\\infty)[\/latex].<\/li>\r\n\t<li>The natural exponential function is [latex]y=e^x[\/latex] and the natural logarithmic function is [latex]y=\\ln x=\\log_e x[\/latex].<\/li>\r\n\t<li>Given an exponential function or logarithmic function in base [latex]a[\/latex], we can make a change of base to convert this function to any base [latex]b&gt;0, \\, b \\ne 1[\/latex]. We typically convert to base [latex]e[\/latex].<\/li>\r\n\t<li>The hyperbolic functions involve combinations of the exponential functions [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1170572452321\">\r\n\t<li><strong>Generalized sine function<\/strong><br \/>\r\n[latex]f(x)=A\\sin(B(x-\\alpha))+C[\/latex]<\/li>\r\n\t<li><strong>Inverse functions<\/strong><br \/>\r\n[latex]f^{-1}(f(x))=x[\/latex]\u00a0 for all\u00a0 [latex]x[\/latex]\u00a0 in\u00a0 [latex]D[\/latex], and\u00a0 [latex]f(f^{-1}(y))=y[\/latex]\u00a0 for all\u00a0 [latex]y[\/latex]\u00a0 in\u00a0 [latex]R[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1170572544592\" class=\"definition\">\r\n<dt>base<\/dt>\r\n<dd id=\"fs-id1170572544597\">the number [latex]b[\/latex] in the exponential function [latex]f(x)=b^x[\/latex] and the logarithmic function [latex]f(x)=\\log_b x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>exponent<\/dt>\r\n<dd id=\"fs-id1170572544654\">the value [latex]x[\/latex] in the expression [latex]b^x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\nhorizontal line test<\/dt>\r\n<dd id=\"fs-id1170572229174\">a function [latex]f[\/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[\/latex], at most, once<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572229190\" class=\"definition\">\r\n<dt>hyperbolic functions<\/dt>\r\n<dd id=\"fs-id1170572544676\">the functions denoted [latex]\\sinh, \\, \\cosh, \\, \\tanh, \\, \\text{csch}, \\, \\text{sech}[\/latex], and [latex]\\coth[\/latex], which involve certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\ninverse function<\/dt>\r\n<dd id=\"fs-id1170572229195\">for a function [latex]f[\/latex], the inverse function [latex]f^{-1}[\/latex] satisfies [latex]f^{-1}(y)=x[\/latex] if [latex]f(x)=y[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294452\" class=\"definition\">\r\n<dt>inverse hyperbolic functions<\/dt>\r\n<dd id=\"fs-id1170572294458\">the inverses of the hyperbolic functions where [latex]\\cosh[\/latex] and [latex]\\text{sech}[\/latex] are restricted to the domain [latex][0,\\infty)[\/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\ninverse trigonometric functions<\/dt>\r\n<dd id=\"fs-id1170572482614\">the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294493\" class=\"definition\">\r\n<dt>natural exponential function<\/dt>\r\n<dd id=\"fs-id1170572294499\">the function [latex]f(x)=e^x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294522\" class=\"definition\">\r\n<dt>natural logarithm<\/dt>\r\n<dd id=\"fs-id1170572294527\">the function [latex]\\ln x=\\log_e x[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572294549\" class=\"definition\">\r\n<dt>number e<\/dt>\r\n<dd id=\"fs-id1170572294554\">as [latex]m[\/latex] gets larger, the quantity [latex](1+(1\/m))^m[\/latex] gets closer to some real number; we define that real number to be [latex]e[\/latex]; the value of [latex]e[\/latex] is approximately 2.718282<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572544649\" class=\"definition\">\r\n<dt>\r\none-to-one function<\/dt>\r\n<dd id=\"fs-id1170572482624\">a function [latex]f[\/latex] is one-to-one if [latex]f(x_1) \\ne f(x_2)[\/latex] if [latex]x_1 \\ne x_2[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572482683\" class=\"definition\">\r\n<dt>restricted domain<\/dt>\r\n<dd id=\"fs-id1170572482689\">a subset of the domain of a function [latex]f[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572481704\" class=\"definition\">\r\n<dt>periodic function<\/dt>\r\n<dd id=\"fs-id1170572481709\">a function is periodic if it has a repeating pattern as the values of [latex]x[\/latex] move from left to right<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572481719\" class=\"definition\">\r\n<dt>radians<\/dt>\r\n<dd id=\"fs-id1170572481724\">for a circular arc of length [latex]s[\/latex] on a circle of radius 1, the radian measure of the associated angle [latex]\\theta [\/latex] is [latex]s[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572552272\" class=\"definition\">\r\n<dt>trigonometric functions<\/dt>\r\n<dd id=\"fs-id1170572552277\">functions of an angle defined as ratios of the lengths of the sides of a right triangle<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1170572552282\" class=\"definition\">\r\n<dt>trigonometric identity<\/dt>\r\n<dd id=\"fs-id1170572552288\">an equation involving trigonometric functions that is true for all angles [latex]\\theta [\/latex] for which the functions in the equation are defined<\/dd>\r\n<\/dl>\r\n<h2>Study Tips<\/h2>\r\n<p><strong>Degrees versus Radians<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize key angle conversions: [latex]30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0, 180\u00b0[\/latex] and their radian equivalents.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize radian measures on a unit circle to understand their relationship to [latex]\u03c0[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When solving problems, check if the angle measure is in degrees or radians before proceeding.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">In calculus and advanced mathematics, assume angles are in radians unless specified otherwise.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use exact values with [latex]\u03c0[\/latex] for precision in radian measures, rather than decimal approximations.<\/li>\r\n<\/ul>\r\n<p><strong>The Six Basic Trigonometric Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice identifying the opposite, adjacent, and hypotenuse sides for different angles in right triangles.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the definitions of all six functions and their reciprocal relationships.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use SOH-CAH-TOA to remember sine, cosine, and tangent ratios.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When solving problems, draw and label a right triangle diagram if one isn't provided.<\/li>\r\n<\/ul>\r\n<p><strong>Trigonometric Identities<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize fundamental identities through regular practice and flashcards. It might be helpful to create visual aids or mnemonics to remember relationships between functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice deriving less common identities from the fundamental ones.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When verifying, start with the more complex side of the equation.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Look for opportunities to factor, square binomials, or use Pythagorean identities.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">In complex problems, try substituting trigonometric expressions with variables to simplify.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Always verify solutions in trigonometric equations by plugging them back into the original equation.<\/li>\r\n<\/ul>\r\n<p><strong>Graphs and Periods of the Trigonometric Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the basic shapes and periods of all six trigonometric functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice sketching transformed graphs by applying one transformation at a time.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]B[\/latex] inside the function affects the period inversely (larger [latex]B[\/latex], shorter period).<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When analyzing a transformed function, identify each component ([latex]A, B, \u03b1, C[\/latex]) and its effect.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use technology to verify your hand-drawn graphs and build intuition about transformations.<\/li>\r\n<\/ul>\r\n<p><strong>Inverse Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Practice finding inverses algebraically by switching [latex]x[\/latex] and [latex]y[\/latex], then solving for [latex]y[\/latex].<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Visualize inverse functions as reflections over [latex]y = x[\/latex] to understand their relationship.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that not all functions have inverses; only one-to-one functions do.\u00a0Use the horizontal line test to quickly determine if a function has an inverse.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When graphing inverses, pay attention to how the domain and range swap.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Be careful not to confuse [latex]f^{-1}(x)[\/latex] with [latex]1\/f(x)[\/latex].<\/li>\r\n<\/ul>\r\n<p><strong>Finding a Function\u2019s Inverse<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Remember to check if a function is one-to-one before attempting to find its inverse.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When swapping x and y, be careful to replace all instances of the variable.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Always verify your inverse function by composing it with the original function.<\/li>\r\n<\/ul>\r\n<p><strong>Graphing Inverse Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">When restricting domains, consider the function's behavior and choose intervals that ensure one-to-one correspondence.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that different domain restrictions can lead to different inverse functions for the same original function.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Always verify that your restricted function is one-to-one using the horizontal line test.<\/li>\r\n<\/ul>\r\n<p><strong>Inverse Trigonometric Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the domains and ranges of inverse trig functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When composing trig and inverse trig functions, pay attention to domain restrictions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the unit circle to understand the relationships between trig functions and their inverses.<\/li>\r\n<\/ul>\r\n<p><strong>Exponential Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the laws of exponents and practice applying them to simplify expressions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare exponential and power functions graphically to understand their differences.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When simplifying complex exponential expressions, break them down step-by-step.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that negative exponents in the denominator can be moved to the numerator with a positive exponent.<\/li>\r\n<\/ul>\r\n<p><strong>Logarithmic Functions<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the properties of logarithms and practice applying them.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the change-of-base formula to evaluate logarithms with uncommon bases.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Practice solving equations that combine exponential and logarithmic functions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Remember that [latex]\\log_b(b^x) = x[\/latex] and [latex]b^{\\log_b(x)} = x[\/latex] for any positive base [latex]b \\neq 1[\/latex].<\/li>\r\n<\/ul>\r\n<p><strong>Hyperbolic Functions\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Memorize the basic hyperbolic identities and practice applying them.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Compare hyperbolic functions to their trigonometric counterparts to understand similarities and differences.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">When working with inverse hyperbolic functions, pay attention to domain restrictions.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use the relationship to exponential functions to help evaluate hyperbolic expressions.<\/li>\r\n<\/ul>","rendered":"<p style=\"text-align: right;\"><a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+1+2024+Build\/Cheat+Sheets\/Calculus+1+Cheat+Sheet_+More+Basic+Functions+and+Graphs.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"font-size: 14pt;\">Download a PDF of this page here.<\/span><\/a><\/p>\n<p style=\"text-align: right;\"><span style=\"font-size: 14pt;\">Download the Spanish version here.<\/span><\/p>\n<h2>Essential Concepts<\/h2>\n<p><strong>Trigonometric Functions<\/strong><\/p>\n<ul id=\"fs-id1170572167654\">\n<li>Radian measure is defined such that the angle associated with the arc of length 1 on the unit circle has radian measure 1. An angle with a degree measure of 180\u00b0 has a radian measure of [latex]\\pi[\/latex] rad.<\/li>\n<li>For acute angles [latex]\\theta[\/latex], the values of the trigonometric functions are defined as ratios of two sides of a right triangle in which one of the acute angles is [latex]\\theta[\/latex].<\/li>\n<li>For a general angle [latex]\\theta[\/latex], let [latex](x,y)[\/latex] be a point on a circle of radius [latex]r[\/latex] corresponding to this angle [latex]\\theta[\/latex]. The trigonometric functions can be written as ratios involving [latex]x, \\, y[\/latex], and [latex]r[\/latex].<\/li>\n<li>The trigonometric functions are periodic. The sine, cosine, secant, and cosecant functions have period [latex]2\\pi[\/latex]. The tangent and cotangent functions have period [latex]\\pi[\/latex].<\/li>\n<\/ul>\n<p><strong>Inverse Functions<\/strong><\/p>\n<ul id=\"fs-id1170572470433\">\n<li>For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.<\/li>\n<li>If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.<\/li>\n<li>For a function [latex]f[\/latex] and its inverse [latex]f^{-1}, \\, f(f^{-1}(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f^{-1}[\/latex] and [latex]f^{-1}(f(x))=x[\/latex] for all [latex]x[\/latex] in the domain of [latex]f[\/latex].<\/li>\n<li>Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.<\/li>\n<li>The graph of a function [latex]f[\/latex] and its inverse [latex]f^{-1}[\/latex] are symmetric about the line [latex]y=x[\/latex].<\/li>\n<\/ul>\n<p><strong>Exponential and Logarithmic Functions<\/strong><\/p>\n<ul id=\"fs-id1170572216270\">\n<li>The exponential function [latex]y=b^x[\/latex] is increasing if [latex]b>1[\/latex] and decreasing if [latex]0<b<1[\/latex]. Its domain is [latex](\u2212\\infty ,\\infty)[\/latex] and its range is [latex](0,\\infty)[\/latex].<\/li>\n<li>The logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of [latex]y=b^x[\/latex]. Its domain is [latex](0,\\infty)[\/latex] and its range is [latex](\u2212\\infty,\\infty)[\/latex].<\/li>\n<li>The natural exponential function is [latex]y=e^x[\/latex] and the natural logarithmic function is [latex]y=\\ln x=\\log_e x[\/latex].<\/li>\n<li>Given an exponential function or logarithmic function in base [latex]a[\/latex], we can make a change of base to convert this function to any base [latex]b>0, \\, b \\ne 1[\/latex]. We typically convert to base [latex]e[\/latex].<\/li>\n<li>The hyperbolic functions involve combinations of the exponential functions [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]. As a result, the inverse hyperbolic functions involve the natural logarithm.<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1170572452321\">\n<li><strong>Generalized sine function<\/strong><br \/>\n[latex]f(x)=A\\sin(B(x-\\alpha))+C[\/latex]<\/li>\n<li><strong>Inverse functions<\/strong><br \/>\n[latex]f^{-1}(f(x))=x[\/latex]\u00a0 for all\u00a0 [latex]x[\/latex]\u00a0 in\u00a0 [latex]D[\/latex], and\u00a0 [latex]f(f^{-1}(y))=y[\/latex]\u00a0 for all\u00a0 [latex]y[\/latex]\u00a0 in\u00a0 [latex]R[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1170572544592\" class=\"definition\">\n<dt>base<\/dt>\n<dd id=\"fs-id1170572544597\">the number [latex]b[\/latex] in the exponential function [latex]f(x)=b^x[\/latex] and the logarithmic function [latex]f(x)=\\log_b x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572544649\" class=\"definition\">\n<dt>exponent<\/dt>\n<dd id=\"fs-id1170572544654\">the value [latex]x[\/latex] in the expression [latex]b^x[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\nhorizontal line test<\/dt>\n<dd id=\"fs-id1170572229174\">a function [latex]f[\/latex] is one-to-one if and only if every horizontal line intersects the graph of [latex]f[\/latex], at most, once<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572229190\" class=\"definition\">\n<dt>hyperbolic functions<\/dt>\n<dd id=\"fs-id1170572544676\">the functions denoted [latex]\\sinh, \\, \\cosh, \\, \\tanh, \\, \\text{csch}, \\, \\text{sech}[\/latex], and [latex]\\coth[\/latex], which involve certain combinations of [latex]e^x[\/latex] and [latex]e^{\u2212x}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\ninverse function<\/dt>\n<dd id=\"fs-id1170572229195\">for a function [latex]f[\/latex], the inverse function [latex]f^{-1}[\/latex] satisfies [latex]f^{-1}(y)=x[\/latex] if [latex]f(x)=y[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294452\" class=\"definition\">\n<dt>inverse hyperbolic functions<\/dt>\n<dd id=\"fs-id1170572294458\">the inverses of the hyperbolic functions where [latex]\\cosh[\/latex] and [latex]\\text{sech}[\/latex] are restricted to the domain [latex][0,\\infty)[\/latex]; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\ninverse trigonometric functions<\/dt>\n<dd id=\"fs-id1170572482614\">the inverses of the trigonometric functions are defined on restricted domains where they are one-to-one functions<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294493\" class=\"definition\">\n<dt>natural exponential function<\/dt>\n<dd id=\"fs-id1170572294499\">the function [latex]f(x)=e^x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294522\" class=\"definition\">\n<dt>natural logarithm<\/dt>\n<dd id=\"fs-id1170572294527\">the function [latex]\\ln x=\\log_e x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572294549\" class=\"definition\">\n<dt>number e<\/dt>\n<dd id=\"fs-id1170572294554\">as [latex]m[\/latex] gets larger, the quantity [latex](1+(1\/m))^m[\/latex] gets closer to some real number; we define that real number to be [latex]e[\/latex]; the value of [latex]e[\/latex] is approximately 2.718282<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>\none-to-one function<\/dt>\n<dd id=\"fs-id1170572482624\">a function [latex]f[\/latex] is one-to-one if [latex]f(x_1) \\ne f(x_2)[\/latex] if [latex]x_1 \\ne x_2[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572482683\" class=\"definition\">\n<dt>restricted domain<\/dt>\n<dd id=\"fs-id1170572482689\">a subset of the domain of a function [latex]f[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572481704\" class=\"definition\">\n<dt>periodic function<\/dt>\n<dd id=\"fs-id1170572481709\">a function is periodic if it has a repeating pattern as the values of [latex]x[\/latex] move from left to right<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572481719\" class=\"definition\">\n<dt>radians<\/dt>\n<dd id=\"fs-id1170572481724\">for a circular arc of length [latex]s[\/latex] on a circle of radius 1, the radian measure of the associated angle [latex]\\theta[\/latex] is [latex]s[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572552272\" class=\"definition\">\n<dt>trigonometric functions<\/dt>\n<dd id=\"fs-id1170572552277\">functions of an angle defined as ratios of the lengths of the sides of a right triangle<\/dd>\n<\/dl>\n<dl id=\"fs-id1170572552282\" class=\"definition\">\n<dt>trigonometric identity<\/dt>\n<dd id=\"fs-id1170572552288\">an equation involving trigonometric functions that is true for all angles [latex]\\theta[\/latex] for which the functions in the equation are defined<\/dd>\n<\/dl>\n<h2>Study Tips<\/h2>\n<p><strong>Degrees versus Radians<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize key angle conversions: [latex]30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0, 180\u00b0[\/latex] and their radian equivalents.<\/li>\n<li class=\"whitespace-normal break-words\">Visualize radian measures on a unit circle to understand their relationship to [latex]\u03c0[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">When solving problems, check if the angle measure is in degrees or radians before proceeding.<\/li>\n<li class=\"whitespace-normal break-words\">In calculus and advanced mathematics, assume angles are in radians unless specified otherwise.<\/li>\n<li class=\"whitespace-normal break-words\">Use exact values with [latex]\u03c0[\/latex] for precision in radian measures, rather than decimal approximations.<\/li>\n<\/ul>\n<p><strong>The Six Basic Trigonometric Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice identifying the opposite, adjacent, and hypotenuse sides for different angles in right triangles.<\/li>\n<li class=\"whitespace-normal break-words\">Memorize the definitions of all six functions and their reciprocal relationships.<\/li>\n<li class=\"whitespace-normal break-words\">Use SOH-CAH-TOA to remember sine, cosine, and tangent ratios.<\/li>\n<li class=\"whitespace-normal break-words\">When solving problems, draw and label a right triangle diagram if one isn&#8217;t provided.<\/li>\n<\/ul>\n<p><strong>Trigonometric Identities<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize fundamental identities through regular practice and flashcards. It might be helpful to create visual aids or mnemonics to remember relationships between functions.<\/li>\n<li class=\"whitespace-normal break-words\">Practice deriving less common identities from the fundamental ones.<\/li>\n<li class=\"whitespace-normal break-words\">When verifying, start with the more complex side of the equation.<\/li>\n<li class=\"whitespace-normal break-words\">Look for opportunities to factor, square binomials, or use Pythagorean identities.<\/li>\n<li class=\"whitespace-normal break-words\">In complex problems, try substituting trigonometric expressions with variables to simplify.<\/li>\n<li class=\"whitespace-normal break-words\">Always verify solutions in trigonometric equations by plugging them back into the original equation.<\/li>\n<\/ul>\n<p><strong>Graphs and Periods of the Trigonometric Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the basic shapes and periods of all six trigonometric functions.<\/li>\n<li class=\"whitespace-normal break-words\">Practice sketching transformed graphs by applying one transformation at a time.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]B[\/latex] inside the function affects the period inversely (larger [latex]B[\/latex], shorter period).<\/li>\n<li class=\"whitespace-normal break-words\">When analyzing a transformed function, identify each component ([latex]A, B, \u03b1, C[\/latex]) and its effect.<\/li>\n<li class=\"whitespace-normal break-words\">Use technology to verify your hand-drawn graphs and build intuition about transformations.<\/li>\n<\/ul>\n<p><strong>Inverse Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Practice finding inverses algebraically by switching [latex]x[\/latex] and [latex]y[\/latex], then solving for [latex]y[\/latex].<\/li>\n<li class=\"whitespace-normal break-words\">Visualize inverse functions as reflections over [latex]y = x[\/latex] to understand their relationship.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that not all functions have inverses; only one-to-one functions do.\u00a0Use the horizontal line test to quickly determine if a function has an inverse.<\/li>\n<li class=\"whitespace-normal break-words\">When graphing inverses, pay attention to how the domain and range swap.<\/li>\n<li class=\"whitespace-normal break-words\">Be careful not to confuse [latex]f^{-1}(x)[\/latex] with [latex]1\/f(x)[\/latex].<\/li>\n<\/ul>\n<p><strong>Finding a Function\u2019s Inverse<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Remember to check if a function is one-to-one before attempting to find its inverse.<\/li>\n<li class=\"whitespace-normal break-words\">When swapping x and y, be careful to replace all instances of the variable.<\/li>\n<li class=\"whitespace-normal break-words\">Always verify your inverse function by composing it with the original function.<\/li>\n<\/ul>\n<p><strong>Graphing Inverse Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">When restricting domains, consider the function&#8217;s behavior and choose intervals that ensure one-to-one correspondence.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that different domain restrictions can lead to different inverse functions for the same original function.<\/li>\n<li class=\"whitespace-normal break-words\">Always verify that your restricted function is one-to-one using the horizontal line test.<\/li>\n<\/ul>\n<p><strong>Inverse Trigonometric Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the domains and ranges of inverse trig functions.<\/li>\n<li class=\"whitespace-normal break-words\">When composing trig and inverse trig functions, pay attention to domain restrictions.<\/li>\n<li class=\"whitespace-normal break-words\">Use the unit circle to understand the relationships between trig functions and their inverses.<\/li>\n<\/ul>\n<p><strong>Exponential Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the laws of exponents and practice applying them to simplify expressions.<\/li>\n<li class=\"whitespace-normal break-words\">Compare exponential and power functions graphically to understand their differences.<\/li>\n<li class=\"whitespace-normal break-words\">When simplifying complex exponential expressions, break them down step-by-step.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that negative exponents in the denominator can be moved to the numerator with a positive exponent.<\/li>\n<\/ul>\n<p><strong>Logarithmic Functions<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the properties of logarithms and practice applying them.<\/li>\n<li class=\"whitespace-normal break-words\">Use the change-of-base formula to evaluate logarithms with uncommon bases.<\/li>\n<li class=\"whitespace-normal break-words\">Practice solving equations that combine exponential and logarithmic functions.<\/li>\n<li class=\"whitespace-normal break-words\">Remember that [latex]\\log_b(b^x) = x[\/latex] and [latex]b^{\\log_b(x)} = x[\/latex] for any positive base [latex]b \\neq 1[\/latex].<\/li>\n<\/ul>\n<p><strong>Hyperbolic Functions\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Memorize the basic hyperbolic identities and practice applying them.<\/li>\n<li class=\"whitespace-normal break-words\">Compare hyperbolic functions to their trigonometric counterparts to understand similarities and differences.<\/li>\n<li class=\"whitespace-normal break-words\">When working with inverse hyperbolic functions, pay attention to domain restrictions.<\/li>\n<li class=\"whitespace-normal break-words\">Use the relationship to exponential functions to help evaluate hyperbolic expressions.<\/li>\n<\/ul>\n","protected":false},"author":15,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":759,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3587"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3587\/revisions"}],"predecessor-version":[{"id":4072,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3587\/revisions\/4072"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3587\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3587"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3587"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3587"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3587"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}