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Essential Concepts

Review of Functions

• A function is a mapping from a set of inputs to a set of outputs with exactly one output for each input.
• If no domain is stated for a function [latex]y=f(x)[/latex], the domain is considered to be the set of all real numbers [latex]x[/latex] for which the function is defined.
• When sketching the graph of a function [latex]f[/latex], each vertical line may intersect the graph, at most, once.
• A function may have any number of zeros, but it has, at most, one [latex]y[/latex]-intercept.
• To define the composition [latex]g\circ f[/latex], the range of [latex]f[/latex] must be contained in the domain of [latex]g[/latex].
• Even functions are symmetric about the [latex]y[/latex]-axis whereas odd functions are symmetric about the origin.

Basic Classes of Functions

• The power function [latex]f(x)=x^n[/latex] is an even function if [latex]n[/latex] is even and [latex]n \ne 0[/latex], and it is an odd function if [latex]n[/latex] is odd.
• The root function [latex]f(x)=x^{1/n}[/latex] has the domain [latex][0,\infty )[/latex] if [latex]n[/latex] is even and the domain [latex](-\infty,\infty )[/latex] if [latex]n[/latex] is odd. If [latex]n[/latex] is odd, then [latex]f(x)=x^{1/n}[/latex] is an odd function.
• The domain of the rational function [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomial functions, is the set of [latex]x[/latex] such that [latex]q(x) \ne 0[/latex].
• Functions that involve the basic operations of addition, subtraction, multiplication, division, and powers are algebraic functions. All other functions are transcendental. Trigonometric, exponential, and logarithmic functions are examples of transcendental functions.
• A polynomial function [latex]f[/latex] with degree [latex]n \ge 1[/latex] satisfies [latex]f(x) \to \pm \infty[/latex] as [latex]x \to \pm \infty[/latex]. The sign of the output as [latex]x \to \infty[/latex] depends on the sign of the leading coefficient only and on whether [latex]n[/latex] is even or odd.
• Vertical and horizontal shifts, vertical and horizontal scalings, and reflections about the [latex]x[/latex]– and [latex]y[/latex]-axes are examples of transformations of functions.

Key Equations

• Composition of two functions
  [latex](g\circ f)(x)=g(f(x))[/latex]
• Absolute value function
  [latex]f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}[/latex]
• Point-slope equation of a line
  [latex]y-y_1=m(x-x_1)[/latex]
• Slope-intercept form of a line
  [latex]y=mx+b[/latex]
• Standard form of a line
  [latex]ax+by=c[/latex]
• Polynomial function
  [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \ldots +a_1x+a_0[/latex]

Glossary

absolute value function

[latex]f(x) = |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}[/latex]

algebraic function

a function involving any combination of only the basic operations of addition, subtraction, multiplication, division, powers, and roots applied to an input variable [latex]x[/latex]

composite function

given two functions [latex]f[/latex] and [latex]g[/latex], a new function, denoted [latex]g\circ f[/latex], such that [latex](g\circ f)(x)=g(f(x))[/latex]

cubic function

a polynomial of degree [latex]3[/latex]; that is, a function of the form [latex]f(x)=ax^3+bx^2+cx+d[/latex], where [latex]a \ne 0[/latex]

decreasing on the interval [latex]I[/latex]

a function decreasing on the interval [latex]I[/latex] if, for all [latex]x_1, \, x_2\in I, \, f(x_1)\ge f(x_2)[/latex] if [latex]x_1

degree

for a polynomial function, the value of the largest exponent of any term

dependent variable

the output variable for a function

domain

the set of inputs for a function

even function

a function is even if [latex]f(−x)=f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]

function

a set of inputs, a set of outputs, and a rule for mapping each input to exactly one output

graph of a function

the set of points [latex](x,y)[/latex] such that [latex]x[/latex] is in the domain of [latex]f[/latex] and [latex]y=f(x)[/latex]

independent variable

the input variable for a function

linear function

a function that can be written in the form [latex]f(x)=mx+b[/latex]

logarithmic function

a function of the form [latex]f(x)=\log_b(x)[/latex] for some base [latex]b>0, \, b \ne 1[/latex] such that [latex]y=\log_b(x)[/latex] if and only if [latex]b^y=x[/latex]

mathematical model

A method of simulating real-life situations with mathematical equations

odd function

a function is odd if [latex]f(−x)=−f(x)[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]

piecewise-defined function

a function that is defined differently on different parts of its domain

point-slope equation

equation of a linear function indicating its slope and a point on the graph of the function

polynomial function

a function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1}+ \cdots +a_1x+a_0[/latex]

power function

a function of the form [latex]f(x)=x^n[/latex] for any positive integer [latex]n \ge 1[/latex]

quadratic function

a polynomial of degree 2; that is, a function of the form [latex]f(x)=ax^2+bx+c[/latex] where [latex]a \ne 0[/latex]

range

the set of outputs for a function

rational function

a function of the form [latex]f(x)=p(x)/q(x)[/latex], where [latex]p(x)[/latex] and [latex]q(x)[/latex] are polynomials

root function

a function of the form [latex]f(x)=x^{1/n}[/latex] for any integer [latex]n \ge 2[/latex]

slope

the change in [latex]y[/latex] for each unit change in [latex]x[/latex]

slope-intercept form

equation of a linear function indicating its slope and [latex]y[/latex]-intercept

standard form

equation of a linear function with both variable terms set equal to a constant, [latex]ax+by=c[/latex].

symmetry about the origin

the graph of a function [latex]f[/latex] is symmetric about the origin if [latex](−x,−y)[/latex] is on the graph of [latex]f[/latex] whenever [latex](x,y)[/latex] is on the graph

symmetry about the [latex]y[/latex]-axis

the graph of a function [latex]f[/latex] is symmetric about the [latex]y[/latex]-axis if [latex](−x,y)[/latex] is on the graph of [latex]f[/latex] whenever [latex](x,y)[/latex] is on the graph

table of values

a table containing a list of inputs and their corresponding outputs

transcendental function

a function that cannot be expressed by a combination of basic arithmetic operations

transformation of a function

a shift, scaling, or reflection of a function

vertical line test

given the graph of a function, every vertical line intersects the graph, at most, once

zeros of a function

when a real number [latex]x[/latex] is a zero of a function [latex]f[/latex], [latex]f(x)=0[/latex]

Study Tips

Functions

• Practice evaluating functions by substituting given values for [latex]x[/latex] in the function’s formula.
• When working with tables, pay attention to how the output changes as the input changes.
• When looking at graphs, observe the overall behavior of the function, including any increases, decreases, or constant sections.
• For algebraic formulas, practice creating tables of values to help visualize the function.

The Domain and Range of a Function

• When determining the domain, look for potential mathematical restrictions like division by zero or even roots of negative numbers.
• For the range, consider the nature of the function (linear, quadratic, etc.) and any limitations on the output values.
• Practice using both set notation and interval notation to express domains and ranges.
• Remember that infinity ([latex]\infty[/latex]) is not a real number but a concept used to describe unbounded sets.
• When working with rational functions, pay special attention to values that make the denominator zero.
• For functions involving square roots, ensure the expression under the root is non-negative.
• Visualize the function graph to help identify the domain and range, especially for common function types.

Intercepts of a Function

• To find [latex]x[/latex]-intercepts, set the function equal to zero and solve for [latex]x[/latex].
• To find the [latex]y[/latex]-intercept, evaluate the function at [latex]x = 0[/latex].
• Remember that [latex]x[/latex]-intercepts are in the form ([latex]x, 0[/latex]), while the [latex]y[/latex]-intercept is in the form ([latex]0, y[/latex]).
• Remember that not all functions have x-intercepts, but most will have a [latex]y[/latex]-intercept.

Symmetry of Functions 

• To check for symmetry algebraically:
  • Even functions: Replace [latex]x[/latex] with [latex]-x[/latex] and see if the result is identical
  • Odd functions: Replace [latex]x[/latex] with [latex]-x[/latex] and see if the result is the negative of the original function
• Visualize symmetry:
  • Even functions: Mirror image across [latex]y[/latex]-axis
  • Odd functions: [latex]180°[/latex] rotation around the origin
• For absolute value functions:
  • Remember they create a “V” shape on the graph
  • The output is always non-negative
  • Transformations can shift or stretch the basic [latex]|x|[/latex] function
• Practice identifying symmetry in various function types (polynomials, rationals, etc.)
• When solving absolute value equations, consider both positive and negative cases

Composing Functions 

• When combining functions, treat each function as a unit and apply the operation.
• For composite functions, work from the inside out: first apply the inner function, then the outer function.
• Pay attention to the domains when combining or composing functions, especially for quotients and composites.
• Practice identifying the difference between [latex](f ∘ g)(x)[/latex] and [latex](g ∘ f)(x)[/latex].
• When working with composing polynomials:
  • Add/subtract: Combine like terms
  • Multiply: Use the FOIL method for binomials or distribute each term

Polynomial Functions 

• Always arrange polynomials in descending order of degree for easy identification of the leading term and degree.
• Remember that the degree is determined by the highest power of the variable, not the largest number in the function.
• The leading coefficient can be positive or negative, and it’s always the number in front of the highest power term.
• When the leading coefficient is [latex]1[/latex] or [latex]-1[/latex], it’s often not written explicitly. For example, [latex]x³ + 2x²[/latex] has a leading coefficient of [latex]1[/latex].
• Be careful with negative exponents or fractional exponents – these are not considered polynomials.

Zeros of Polynomial Functions

• For quadratic functions, try factoring first if the coefficients are simple.
• Memorize the quadratic formula and practice using it.
• Pay attention to the sign of the discriminant to predict the nature of roots before solving.
• When using the quadratic formula, simplify under the square root if possible before calculating further.
• Double-check your solutions by plugging them back into the original equation.
• Remember that the number of real zeros a polynomial can have is at most equal to its degree.

Graphs of Polynomial Functions Basics

• Always identify the degree and the sign of the leading coefficient first.
• Remember: Even degree = same direction ends, Odd degree = opposite direction ends.
• Practice sketching basic shapes for different degrees and leading coefficient signs.
• For odd-degree polynomials, a negative input will always yield a negative output (assuming positive leading coefficient).
• Look at the behavior of the graph as [latex]x[/latex] approaches positive and negative infinity to determine the degree’s parity and leading coefficient’s sign.
• Compare graphs of polynomials with the same degree but different leading coefficient signs to see the reflection effect.
• When in doubt, plot a few key points to confirm the graph’s behavior.

Piecewise-Defined Functions

• Clearly identify the domain for each piece of the function.
• When graphing, use open or closed circles at endpoints to show inclusion or exclusion of boundary points.
• Check for continuity at the points where the function definition changes.
• When solving equations involving piecewise functions, consider each piece separately.
• Use technology to help visualize complex piecewise functions.

Algebraic Functions and Transcendental Functions

• To identify algebraic functions, look for operations limited to addition, subtraction, multiplication, division, and rational exponents.
• Remember that any finite combination of algebraic functions is still algebraic.
• Transcendental functions often involve trigonometric ratios, e, pi, or logarithms.
• Practice identifying functions by their form:
  • Fraction of polynomials? Rational function.
  • Variable under a root? Root function.
  • Variable in the exponent? Likely exponential (transcendental).
• Be cautious with composite functions – they may combine algebraic and transcendental elements.
• When in doubt, ask yourself: “Can this be broken down into basic algebraic operations?”

Transformations of Functions

• Practice identifying transformations from equations.
• Sketch basic function graphs and apply transformations step-by-step.
• Remember that horizontal transformations work in the “opposite” direction of what you might expect.
• Use the order of operations for multiple transformations: parentheses, exponents, multiplication/division, addition/subtraction.
• Familiarize yourself with basic toolkit functions as starting points for transformations.
• When dealing with complex transformations, break them down into individual steps.
• Use technology to verify your hand-drawn transformed graphs.