{"id":479,"date":"2025-07-10T17:06:38","date_gmt":"2025-07-10T17:06:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=479"},"modified":"2025-12-17T16:32:11","modified_gmt":"2025-12-17T16:32:11","slug":"functions-and-function-notation-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/functions-and-function-notation-learn-it-2\/","title":{"raw":"Functions and Function Notation: Learn It 3","rendered":"Functions and Function Notation: Learn It 3"},"content":{"raw":"
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Using Function Notation<\/h2>\r\n<\/div>\r\n<\/div>\r\n
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Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions.<\/p>\r\n\r\n

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function notation<\/h3>\r\nThe notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as \"[latex]y[\/latex] is a function of [latex]x[\/latex].\" The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex], or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.\r\n\r\n<\/section>\r\n

To represent \"height is a function of age,\" we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age. The letters [latex]f,g[\/latex], and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,y[\/latex], and [latex]z[\/latex] to represent numbers and [latex]A,B[\/latex],\u00a0and [latex]C[\/latex] to represent sets.<\/p>\r\n\r\n

[latex]\\begin{gathered}\\begin{cases}\\begin{align}&h\\text{ is }f\\text{ of }a && \\text{We name the function }f;\\text{ height is a function of age}. \\\\ &h=f\\left(a\\right) && \\text{We use parentheses to indicate the function input}\\text{. } \\\\ &f\\left(a\\right) && \\text{We name the function }f;\\text{ the expression is read as \"}f\\text{ of }a\\text{.\"} \\end{align} \\end{cases} \\\\{ } \\end{gathered}[\/latex]<\/div>\r\n

Remember, we can use any letter to name the function; the notation [latex]h\\left(a\\right)[\/latex] shows us that [latex]h[\/latex] depends on [latex]a[\/latex]. The value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\r\n

We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means \"first add a<\/em> and b<\/em>, and the result is the input for the function f<\/em>.\" The operations must be performed in this order to obtain the correct result.<\/p>\r\n\r\n

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Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.<\/p>\r\n[reveal-answer q=\"413960\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"413960\"]\r\n\r\nThe number of days in a month is a function of the name of the month, so if we name the function [latex]f[\/latex], we write [latex]\\text{days}=f\\left(\\text{month}\\right)[\/latex]\u00a0or [latex]d=f\\left(m\\right)[\/latex]. The name of the month is the input to a \"rule\" that associates a specific number (the output) with each input.\r\n\r\n\"The\r\n\r\n \r\n

For example, [latex]f\\left(\\text{March}\\right)=31[\/latex], because March has 31 days. The notation [latex]d=f\\left(m\\right)[\/latex] reminds us that the number of days, [latex]d[\/latex] (the output), is dependent on the name of the month, [latex]m[\/latex] (the input).<\/p>\r\n\r\n

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Analysis of the Solution<\/h4>\r\n

Note that the inputs to a function do not have to be numbers; function inputs can be names of people, labels of geometric objects, or any other element that determines some kind of output. However, most of the functions we will work with in this book will have numbers as inputs and outputs.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>

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A function [latex]N=f\\left(y\\right)[\/latex] gives the number of police officers, [latex]N[\/latex], in a town in year [latex]y[\/latex]. What does [latex]f\\left(2005\\right)=300[\/latex] represent?<\/p>\r\n[reveal-answer q=\"17695\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"17695\"]\r\n\r\nWhen we read [latex]f\\left(2005\\right)=300[\/latex], we see that the input year is 2005. The value for the output, the number of police officers [latex]\\left(N\\right)[\/latex], is 300. Remember, [latex]N=f\\left(y\\right)[\/latex]. The statement [latex]f\\left(2005\\right)=300[\/latex] tells us that in the year 2005 there were 300 police officers in the town.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>

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When we know an input value and want to determine the corresponding output value for a function, we evaluate<\/em> the function. Evaluating will always produce one result because each input value of a function corresponds to exactly one output value.<\/p>\r\n

When we know an output value and want to determine the input values that would produce that output value, we set the output equal to the function\u2019s formula and solve<\/em> for the input. Solving can produce more than one solution because different input values can produce the same output value.<\/p>\r\n\r\n

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Evaluating Functions in Algebraic Forms<\/h2>\r\n

When we have a function in formula form, it is usually a simple matter to evaluate the function. For example the function [latex]f\\left(x\\right)=5 - 3{x}^{2}[\/latex] can be evaluated by squaring the input value, multiplying by 3, and then subtracting the product from 5.<\/p>\r\n\r\n

Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], evaluate [latex]h\\left(4\\right)[\/latex].[reveal-answer q=\"230509\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"230509\"]To evaluate [latex]h\\left(4\\right)[\/latex], we substitute the value 4 for the input variable [latex]p[\/latex] in the given function.\r\n
[latex]\\begin{align}h\\left(p\\right)&={p}^{2}+2p \\\\ h\\left(4\\right)&={\\left(4\\right)}^{2}+2\\left(4\\right) \\\\ &=16+8 \\\\ &=24 \\end{align}[\/latex]<\/div>\r\n

Therefore, for an input of 4, we have an output of 24.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

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Evaluate [latex]f\\left(x\\right)={x}^{2}+3x - 4[\/latex] at<\/p>\r\n\r\n

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  1. [latex]2[\/latex]<\/li>\r\n \t
  2. [latex]a[\/latex]<\/li>\r\n \t
  3. [latex]a+h[\/latex]<\/li>\r\n \t
  4. [latex]\\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"52497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"52497\"]\r\n

    Replace the [latex]x[\/latex]\u00a0in the function with each specified value.<\/p>\r\n\r\n

      \r\n \t
    1. Because the input value is a number, 2, we can use algebra to simplify.\r\n
      [latex]\\begin{align}f\\left(2\\right)&={2}^{2}+3\\left(2\\right)-4 \\\\ &=4+6 - 4 \\\\ &=6 \\end{align}[\/latex]<\/div><\/li>\r\n \t
    2. In this case, the input value is a letter so we cannot simplify the answer any further.\r\n
      [latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div><\/li>\r\n \t
    3. With an input value of [latex]a+h[\/latex], we must use the distributive property.\r\n
      [latex]\\begin{align}f\\left(a+h\\right)&={\\left(a+h\\right)}^{2}+3\\left(a+h\\right)-4 \\\\ &={a}^{2}+2ah+{h}^{2}+3a+3h - 4 \\end{align}[\/latex]<\/div><\/li>\r\n \t
    4. In this case, we apply the input values to the function more than once, and then perform algebraic operations on the result. We already found that\r\n
      [latex]f\\left(a+h\\right)={a}^{2}+2ah+{h}^{2}+3a+3h - 4[\/latex]<\/div>\r\nand we know that\r\n
      [latex]f\\left(a\\right)={a}^{2}+3a - 4[\/latex]<\/div>\r\n

      Now we combine the results and simplify.<\/p>\r\n\r\n

      [latex]\\begin{align} \\frac{f\\left(a+h\\right)-f\\left(a\\right)}{h}&=\\frac{\\left({a}^{2}+2ah+{h}^{2}+3a+3h - 4\\right)-\\left({a}^{2}+3a - 4\\right)}{h} \\\\[1.5mm]&=\\frac{2ah+{h}^{2}+3h}{h} \\\\[1.5mm]&=\\frac{h\\left(2a+h+3\\right)}{h} &&\\text{Factor out }h. \\\\[1.5mm]&=2a+h+3 && \\text{Simplify}. \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]317458[\/ohm_question]\r\n\r\n<\/section>\r\n
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      Given the function [latex]h\\left(p\\right)={p}^{2}+2p[\/latex], solve for [latex]h\\left(p\\right)=3[\/latex].[reveal-answer q=\"630043\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"630043\"]\r\n

      [latex]\\begin{align}h\\left(p\\right)=3 \\\\ {p}^{2}+2p=3 &\\hspace{3mm} \\text{Substitute the original function }h\\left(p\\right)={p}^{2}+2p. \\\\ {p}^{2}+2p - 3=0 &\\hspace{3mm} \\text{Subtract 3 from each side}. \\\\ \\left(p+3\\text{)(}p - 1\\right)=0 &\\hspace{3mm} \\text{Factor}. \\end{align}[\/latex]<\/p>\r\nIf [latex]\\left(p+3\\right)\\left(p - 1\\right)=0[\/latex], either [latex]\\left(p+3\\right)=0[\/latex] or [latex]\\left(p - 1\\right)=0[\/latex] (or both of them equal 0). We will set each factor equal to 0 and solve for [latex]p[\/latex] in each case.\r\n

      [latex]\\begin{align}\\left(p+3\\right)=0, & \\hspace{3mm} p=-3 \\\\ \\left(p - 1\\right)=0, & \\hspace{3mm} p=1 \\end{align}[\/latex]<\/p>\r\nThis gives us two solutions. The output [latex]h\\left(p\\right)=3[\/latex] when the input is either [latex]p=1[\/latex] or [latex]p=-3[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 5<\/b>[\/caption]\r\n\r\nWe can also verify by graphing as in Figure 5. The graph verifies that [latex]h\\left(1\\right)=h\\left(-3\\right)=3[\/latex] and [latex]h\\left(4\\right)=24[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/div>\r\n

      [ohm_question hide_question_numbers=1]317459[\/ohm_question]\r\n\r\n<\/section><\/div>\r\n

      Evaluating Functions Expressed in Formulas<\/h2>\r\n

      Some functions are defined by mathematical rules or procedures expressed in equation<\/strong> form. If it is possible to express the function output with a formula<\/strong> involving the input quantity, then we can define a function in algebraic form. For example, the equation [latex]2n+6p=12[\/latex] expresses a functional relationship between [latex]n[\/latex] and [latex]p[\/latex]. We can rewrite it to decide if [latex]p[\/latex] is a function of [latex]n[\/latex].<\/p>\r\n\r\n

      Express the relationship [latex]2n+6p=12[\/latex] as a function [latex]p=f\\left(n\\right)[\/latex], if possible.\r\n

      [reveal-answer q=\"65760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"65760\"]<\/p>\r\n

      To express the relationship in this form, we need to be able to write the relationship where [latex]p[\/latex] is a function of [latex]n[\/latex], which means writing it as p = expression involving n.<\/p>\r\n

      [latex]\\begin{align} &2n+6p=12 \\\\ &6p=12 - 2n && \\text{Subtract }2n\\text{ from both sides}. \\\\ &p=\\frac{12 - 2n}{6} && \\text{Divide both sides by 6 and simplify}. \\\\ &p=\\frac{12}{6}-\\frac{2n}{6} \\\\ &p=2-\\frac{1}{3}n \\end{align}[\/latex]<\/p>\r\n

      Therefore, [latex]p[\/latex] as a function of [latex]n[\/latex] is written as<\/p>\r\n

      [latex]p=f\\left(n\\right)=2-\\frac{1}{3}n[\/latex]<\/p>\r\n\r\n

      Analysis of the Solution<\/h4>\r\n

      It is important to note that not every relationship expressed by an equation can also be expressed as a function with a formula.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      Does the equation [latex]{x}^{2}+{y}^{2}=1[\/latex] represent a function with [latex]x[\/latex] as input and [latex]y[\/latex] as output? If so, express the relationship as a function [latex]y=f\\left(x\\right)[\/latex].[reveal-answer q=\"518628\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518628\"]First we subtract [latex]{x}^{2}[\/latex] from both sides.\r\n
      [latex]\\begin{align} &{x}^{2}+{y}^{2}=1 \\\\ &{y}^{2}=1-{x}^{2} && \\text{Subtract } {x}^{2} \\text{ from both sides} \\\\ &y=\\pm \\sqrt{1-{x}^{2}} && \\text{Solve for } y \\text{ using the square root principle.} \\end{align}[\/latex]<\/div>\r\n \r\n\r\nWe get two outputs corresponding to the same input, so this relationship cannot be represented as a single function\r\n

      [latex]y=f\\left(x\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

      [ohm_question hide_question_numbers=1]317460[\/ohm_question]<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"
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      Using Function Notation<\/h2>\n<\/div>\n<\/div>\n
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      \n
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      Once we determine that a relationship is a function, we need to display and define the functional relationships so that we can understand and use them, and sometimes also so that we can program them into computers. There are various ways of representing functions.<\/p>\n

      \n

      function notation<\/h3>\n

      The notation [latex]y=f\\left(x\\right)[\/latex] defines a function named [latex]f[\/latex]. This is read as “[latex]y[\/latex] is a function of [latex]x[\/latex].” The letter [latex]x[\/latex] represents the input value, or independent variable. The letter [latex]y[\/latex], or [latex]f\\left(x\\right)[\/latex], represents the output value, or dependent variable.<\/p>\n<\/section>\n

      To represent “height is a function of age,” we start by identifying the descriptive variables [latex]h[\/latex]\u00a0for height and [latex]a[\/latex]\u00a0for age. The letters [latex]f,g[\/latex], and [latex]h[\/latex] are often used to represent functions just as we use [latex]x,y[\/latex], and [latex]z[\/latex] to represent numbers and [latex]A,B[\/latex],\u00a0and [latex]C[\/latex] to represent sets.<\/p>\n

      [latex]\\begin{gathered}\\begin{cases}\\begin{align}&h\\text{ is }f\\text{ of }a && \\text{We name the function }f;\\text{ height is a function of age}. \\\\ &h=f\\left(a\\right) && \\text{We use parentheses to indicate the function input}\\text{. } \\\\ &f\\left(a\\right) && \\text{We name the function }f;\\text{ the expression is read as \"}f\\text{ of }a\\text{.\"} \\end{align} \\end{cases} \\\\{ } \\end{gathered}[\/latex]<\/div>\n

      Remember, we can use any letter to name the function; the notation [latex]h\\left(a\\right)[\/latex] shows us that [latex]h[\/latex] depends on [latex]a[\/latex]. The value [latex]a[\/latex] must be put into the function [latex]h[\/latex] to get a result. The parentheses indicate that age is input into the function; they do not indicate multiplication.<\/p>\n

      We can also give an algebraic expression as the input to a function. For example [latex]f\\left(a+b\\right)[\/latex] means “first add a<\/em> and b<\/em>, and the result is the input for the function f<\/em>.” The operations must be performed in this order to obtain the correct result.<\/p>\n

      \n

      Use function notation to represent a function whose input is the name of a month and output is the number of days in that month.<\/p>\n