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Mathematics LibreTexts

1.10: Functions (Review)

  • Page ID
    20866
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    1.1: Functions and Function Notation

    For the exercises 1-4, determine whether the relation is a function.

    1) \(\{(a,b),(c,d),(e,d)\}\)

    Answer

    function

    2) \(\{(5,2),(6,1),(6,2),(4,8)\}\)

    3) \(y^2+4=x\),for \(x\) the independent variable and \(y\) the dependent variable

    Answer

    not a function

    4) Is the graph in the Figure below a function?

    CNX_Precalc_Figure_01_07_208.jpg

    For the exercises 5-6, evaluate the function at the indicated values: \(f(-3); f(2); f(-a); -f(a); f(a+h)\)

    5) \(f(x)=-2x^2+3x\)

    Answer

    \(f(-3)=-27; f(2)=-2;f(-a)=-2a^2-3a;-f(a)=2a^2-3a;f(a+h)=-2a^2+3a-4ah+3h-2h^2\)

    6) \(f(x)=2|3x-1|\)

    For the exercises 7-8, determine whether the functions are one-to-one.

    7) \(f(x)=-3 x+5\)

    Answer

    one-to-one

    8) \(f(x)=|x-3|\)

    For the exercises 9-11, use the vertical line test to determine if the relation whose graph is provided is a function.

    9)

    CNX_Precalc_Figure_01_07_209.jpg

    Answer

    function

    10)

    CNX_Precalc_Figure_01_07_210.jpg

    11)

    CNX_Precalc_Figure_01_07_211.jpg

    Answer

    function

    For the exercises 12-13, graph the functions.

    12) \(f(x)=|x+1|\)

    13) \(f(x)=x^{2}-2\)

    Answer

    CNX_Precalc_Figure_01_07_213.jpg

    For the exercises 14-17, use the Figure below to approximate the values.

    CNX_Precalc_Figure_01_07_215.jpg

    14) \(f(2)\)

    15) \(f(-2)\)

    Answer

    \(2\)

    16) If \(f(x)=-2\), then solve for \(x\)

    17) If \(f(x)=1\), then solve for \(x\)

    Answer

    \(x=-1.8\) or \(x=1.8\)

    For the exercises 18-19, use the function \(h(t)=-16 t^{2}+80t\) to find the values.

    18) \(\dfrac{h(2)-h(1)}{2-1}\)

    19) \(\dfrac{h(a)-h(1)}{a-1}\)

    Answer

    \(\dfrac{-64+80 a-16 a^{2}}{-1+a}=-16 a+64\)

    1.2: Domain and Range

    For the exercises 1-4, find the domain of each function, expressing answers using interval notation.

    1) \(f(x)=\dfrac{2}{3 x+2}\)

    2) \(f(x)=\frac{x-3}{x^{2}-4 x-12}\)

    Answer

    \((-\infty,-2) \cup(-2,6) \cup(6, \infty)\)

    3)

    4) Graph this piecewise function: \(f(x)=\left\{\begin{array}{ll}{x+1} & {x<-2} \\ {-2 x-3} & {x \geq-2}\end{array}\right.\)

    Answer

    CNX_Precalc_Figure_01_07_214.jpg

    1.3: Rates of Change and Behavior of Graphs

    For the exercises 1-3, find the average rate of change of the functions from \(x=1\) to \(x=2\)

    1) \(f(x)=4 x-3\)

    2) \(f(x)=10 x^{2}+x\)

    Answer

    \(31\)

    3) \(f(x)=-\dfrac{2}{x^{2}}\)

    For the exercises 4-6, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

    4)

    CNX_Precalc_Figure_01_07_216.jpg

    Answer

    increasing \((2, \infty)\); decreasing \((-\infty, 2)\)

    5)

    CNX_Precalc_Figure_01_07_217.jpg

    6)

    CNX_Precalc_Figure_01_07_218.jpg

    Answer

    increasing \((-3,1)\); constant \((-\infty,-3) \cup(1, \infty)\)

    7) Find the local minimum of the function graphed in Exercise 4.

    8) Find the local extrema for the function graphed in Exercise 5.

    Answer

    local minimum \((-2,-3)\); local maximum \((1,3)\)

    9) For the graph in the Figure in Exercise 10, the domain of the function is \([-3,3]\). The range is \([-10,10]\). Find the absolute minimum of the function on this interval.

    10) Find the absolute maximum of the function graphed in the Figure below.

    CNX_Precalc_Figure_01_07_219.jpg

    Answer

    \((-1.8,10)\)

    1.4: Composition of Functions

    For the exercises 1-5, find \((f \circ g)(x)\) and \((g \circ f)(x)\) for each pair of functions.

    1) \(f(x)=4-x, g(x)=-4x\)

    2) \(f(x)=3 x+2, g(x)=5-6x\)

    Answer

    \((f \circ g)(x)=17-18 x ;(g \circ f)(x)=-7-18x\)

    3) \(f(x)=x^{2}+2 x, g(x)=5 x+1\)

    4) \(f(x)=\sqrt{x+2}, g(x)=\dfrac{1}{x}\)

    Answer

    \((f \circ g)(x)=\sqrt{\dfrac{1}{x}+2} ;(g \circ f)(x)=\dfrac{1}{\sqrt{x+2}}\)

    5) \(f(x)=\dfrac{x+3}{2}, g(x)=\sqrt{1-x}\)

    For the exercises 6-9, find \((f \circ g)\) and the domain for \((f \circ g)(x)\) for each pair of functions.

    6) \(f(x)=\frac{x+1}{x+4}, g(x)=\frac{1}{x}\)

    Answer

    \((f \circ g)(x)=\dfrac{1+x}{1+4 x}, x \neq 0, x \neq-\dfrac{1}{4}\)

    7) \(f(x)=\dfrac{1}{x+3}, g(x)=\dfrac{1}{x-9}\)

    8) \(f(x)=\dfrac{1}{x}, g(x)=\sqrt{x}\)

    Answer

    \((f \circ g)(x)=\frac{1}{\sqrt{x}}, x>0\)

    9) \(f(x)=\frac{1}{x^{2}-1}, g(x)=\sqrt{x+1}\)

    For the exercises 10-11, express each function \(H\) as a composition of two functions \(f\) and \(g\) where \(H(x)=(f \circ g)(x)\)

    10) \(H(x)=\sqrt{\frac{2 x-1}{3 x+4}}\)

    Answer

    sample: \(g(x)=\dfrac{2 x-1}{3 x+4}; f(x)=\sqrt{x}\)

    11) \(H(x)=\dfrac{1}{\left(3 x^{2}-4\right)^{-3}}\)

    1.5: Transformation of Functions

    For the exercises 1-8, sketch a graph of the given function.

    1) \(f(x)=(x-3)^{2}\)

    Answer

    CNX_Precalc_Figure_01_07_220.jpg

    2) \(f(x)=(x+4)^{3}\)

    3) \(f(x)=\sqrt{x}+5\)

    Answer

    CNX_Precalc_Figure_01_07_222.jpg

    4) \(f(x)=-x^{3}\)

    5) \(f(x)=\sqrt[3]{-x}\)

    Answer

    CNX_Precalc_Figure_01_07_224.jpg

    6) \(f(x)=5 \sqrt{-x}-4\)

    7) \(f(x)=4[|x-2|-6]\)

    Answer

    CNX_Precalc_Figure_01_07_226.jpg

    8) \(f(x)=-(x+2)^{2}-1\)

    For the exercises 9-10, sketch the graph of the function \(g\) if the graph of the function \(f\) is shown in the Figure below.

    CNX_Precalc_Figure_01_07_247.jpg

    9) \(g(x)=f(x-1)\)

    Answer

    CNX_Precalc_Figure_01_07_228.jpg

    10) \(g(x)=3 f(x)\)

    For the exercises 11-12, write the equation for the standard function represented by each of the graphs below.

    11)

    CNX_Precalc_Figure_01_07_230.jpg

    Answer

    \(f(x)=|x-3|\)

    12)

    CNX_Precalc_Figure_01_07_231.jpg

    For the exercises 13-15, determine whether each function below is even, odd, or neither.

    13) \(f(x)=3 x^{4}\)

    Answer

    even

    14) \(g(x)=\sqrt{x}\)

    15) \(h(x)=\frac{1}{x}+3 x\)

    Answer

    odd

    For the exercises 16-18, analyze the graph and determine whether the graphed function is even, odd, or neither.

    16)

    CNX_Precalc_Figure_01_07_232.jpg

    17)

    CNX_Precalc_Figure_01_07_233.jpg

    Answer

    even

    18)

    CNX_Precalc_Figure_01_07_234.jpg

    1.6: Absolute Value Functions

    For the exercises 1-3, write an equation for the transformation of \(f(x)=|x|\).

    1)

    CNX_Precalc_Figure_01_07_235.jpg

    Answer

    \(f(x)=\dfrac{1}{2}|x+2|+1\)

    2)

    CNX_Precalc_Figure_01_07_236.jpg

    3)

    CNX_Precalc_Figure_01_07_237.jpg

    Answer

    \(f(x)=-3|x-3|+3\)

    For the exercises 4-6, graph the absolute value function.

    4) \(f(x)=|x-5|\)

    5) \(f(x)=-|x-3|\)

    Answer

    CNX_Precalc_Figure_01_07_239.jpg

    6) \(f(x)=|2 x-4|\)

    For the exercises 7-8, solve the absolute value equation.

    7) \(|x+4|=18\)

    Answer

    \(x=-22, x=14\)

    8) \(\left|\dfrac{1}{3} x+5\right|=\left|\dfrac{3}{4} x-2\right|\)

    For the exercises 9-10, solve the inequality and express the solution using interval notation.

    9) \(|3 x-2|<7\)

    Answer

    \(\left(-\dfrac{5}{3}, 3\right)\)

    10) \(\left|\dfrac{1}{3} x-2\right| \leq 7\)

    1.7: Inverse Functions

    For the exercises 1-2, find \(f^{-1}(x)\) for each function.

    1) \(f(x)=9+10 x\)

    2) \(f(x)=\dfrac{x}{x+2}\)

    Answer

    \(f^{-1}(x)=\dfrac{-2 x}{x-1}\)

    3) For the following exercise, find a domain on which the function \(f\) is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain. \[f(x)=x^{2}+1\]

    4) Given \(f(x)=x^{3}-5\) and \(g(x)=\sqrt[3]{x+5} \) :

    1. Find \(f(g(x))\) and \(g(f(x))\).
    2. What does the answer tell us about the relationship between \(f(x)\) and \(g(x) ?\)
    Answer
    1. \(f(g(x))=x\) and \(g(f(x))=x\)
    2. This tells us that \(f\) and \(g\) are inverse functions

    For the exercises 5-8, use a graphing utility to determine whether each function is one-to-one.

    5) \(f(x)=\dfrac{1}{x}\)

    Answer

    The function is one-to-one.

    CNX_Precalc_Figure_01_07_248.jpg

    6) \(f(x)=-3 x^{2}+x\)

    Answer

    The function is not one-to-one.

    CNX_Precalc_Figure_01_07_249.jpg

    7) If \(f(5)=2,\) find \(f^{-1}(2)\)

    Answer

    \(5\)

    8) If \(f(1)=4,\) find \(f^{-1}(4)\)

    Practice Test

    For the exercises 1-2, determine whether each of the following relations is a function.

    1) \(y=2 x+8\)

    Answer

    The relation is a function.

    2) \(\{(2,1),(3,2),(-1,1),(0,-2)\}\)

    For the exercises 3-4, evaluate the function \(f(x)=-3 x^{2}+2 x\) at the given input.

    3) \(f(-2)\)

    Answer

    \(-16\)

    4) \(f(a)\)

    5) Show that the function \(f(x)=-2(x-1)^{2}+3\) is not one-to-one.

    Answer

    The graph is a parabola and the graph fails the horizontal line test.

    6) Write the domain of the function \(f(x)=\sqrt{3-x}\) in interval notation.

    7) Given \(f(x)=2 x^{2}-5 x,\) find \(f(a+1)-f(1)\)

    Answer

    \(2 a^{2}-a\)

    8) Graph the function \(f(x)=\left\{\begin{array}{ccc}{x+1} & {\text { if }} & {-2<x<3} \\ {-x} & {\text { if }} & {x \geq 3}\end{array}\right.\)

    9) Find the average rate of change of the function \(f(x)=3-2 x^{2}+x\) by finding \(\dfrac{f(b)-f(a)}{b-a}\)

    Answer

    \(-2(a+b)+1\)

    For the exercises 10-11, use the functions \(f(x)=3-2 x^{2}+x\) and \(g(x)=\sqrt{x}\) to find the composite functions.

    10) \((g \circ f)(x)\)

    11) \((g \circ f)(1)\)

    Answer

    \(\sqrt{2}\)

    12) Express \(H(x)=\sqrt[3]{5 x^{2}-3 x}\) a composition of two functions, \(f\) and \(g,\) where \((f \circ g)(x)=H(x)\)

    For the exercises 13-14, graph the functions by translating, stretching, and/or compressing a toolkit function.

    13) \(f(x)=\sqrt{x+6}-1\)

    Answer

    CNX_Precalc_Figure_01_07_242.jpg

    14) \(f(x)=\dfrac{1}{x+2}-1\)

    For the exercises 15-17, determine whether the functions are even, odd, or neither.

    15) \(f(x)=-\dfrac{5}{x^{2}}+9 x^{6}\)

    Answer

    even

    16) \(f(x)=-\dfrac{5}{x^{3}}+9 x^{5}\)

    17) \(f(x)=\dfrac{1}{x}\)

    Answer

    odd

    18) Graph the absolute value function \(f(x)=-2|x-1|+3\).

    19) Solve \(|2 x-3|=17\).

    Answer

    \(x=-7\) and \(x=10\)

    20) Solve \(-\left|\dfrac{1}{3} x-3\right| \geq 17\) . Express the solution in interval notation.

    For the exercises 21-22, find the inverse of the function.

    21) \(f(x)=3 x-5\)

    Answer

    \(f^{-1}(x)=\dfrac{x+5}{3}\)

    22) \(f(x)=\dfrac{4}{x+7}\)

    For the exercises 23-26, use the graph of \(g\) shown in the Figure below.

    23) On what intervals is the function increasing?

    Answer

    \((-\infty,-1.1)\) and \((1.1, \infty)\)

    24) On what intervals is the function decreasing?

    25) Approximate the local minimum of the function. Express the answer as an ordered pair.

    Answer

    \((1.1,-0.9)\)

    26) Approximate the local maximum of the function. Express the answer as an ordered pair.

    For the exercises 27-29, use the graph of the piecewise function shown in the Figure below.

    27) Find \(f(2)\).

    Answer

    \(f(2)=2\)

    28) Find \(f(-2)\).

    29) Write an equation for the piecewise function.

    Answer

    \(f(x)=\left\{\begin{array}{cl}{|x|} & {\text { if } x \leq 2} \\ {3} & {\text { if } x>2}\end{array}\right.\)

    For the exercises 30-35, use the values listed in the Table below.

    \(F(x)\)
    0 1
    1 3
    2 5
    3 7
    4 9
    5 11
    6 13
    7 15
    8 17

    30) Find \(F(6)\).

    31) Solve the equation \(F(x)=5\)

    Answer

    \(x=2\)

    32) Is the graph increasing or decreasing on its domain?

    33) Is the function represented by the graph one-to-one?

    Answer

    yes

    34) Find \(F^{-1}(15)\).

    35) Given \(f(x)=-2 x+11,\) find \(f^{-1}(x)\).

    Answer

    \(f^{-1}(x)=-\dfrac{x-11}{2}\)

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