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Section4.11Additional Practice Related to Functions

Subsection4.11.1Exercises

As directed, determine either the specified function value or determine the solution set to the stated equation.

1

Determine \(k(9)\) where \(k(x)=3+4x-x^2\text{.}\)

Solution

\(k(9)=-42\)

2

Determine \(y(-3)\) where \(y(t)=5-x-x^3\text{.}\)

Solution

\(y(-3)=35\)

3

Determine the solution set to \(m(x)=5\) where \(m(x)=3x-7\text{.}\)

Solution

The solution set is \(\{4\}\text{.}\)

4

Determine the solution set to \(p(t)=-3\) where \(p(t)=t^2-7t-47\text{.}\)

Solution

The solution set is \(\{-4,11\}\text{.}\)

5

Determine \(f(11)\) where \(f(x)=\frac{x-4}{x+3}\text{.}\)

Solution

\(f(11)=\frac{1}{2}\)

6

Determine the solution set to \(w(t)=6\) where \(w(t)=\sqrt{3t-9}\text{.}\)

Solution

The solution set is \(\{15\}\text{.}\)

7

Determine \(w(6)\) where \(w(t)=\sqrt{3t-9}\text{.}\)

Solution

\(w(6)=3\)

8

Determine the solution set to \(q(x)=12\) where \(q(x)=3+\abs{x-1}\text{.}\)

Solution

The solution set is \(\{-8,10\}\text{.}\)

Work with a function presented in graphical form.

9

Determine each of the following function values based upon the function \(k\) shown in FigureĀ 4.11.1.

  1. \(k(2)\)
  2. \(k(0)\)
  3. \(k(4)\)
  4. \(k(-3)\)

Determine the solution set to each of the following equations based upon the function \(k\) shown in FigureĀ 4.11.1.

  1. \(k(x)=1\)
  2. \(k(x)=4\)
  3. \(k(x)=5\)
  4. \(k(x)=6\)
plain text
Figure4.11.1\(y=k(x)\)
Solution
  1. \(k(2)\) is undefined.
  2. \(k(0)=-3\)
  3. \(k(4)=4\)
  4. \(k(-3)=-5\)
  1. The solution set is \(\{-1,5\}\text{.}\)
  2. The solution set is \(\{4\}\text{.}\)
  3. The solution set is \(\{3\}\text{.}\)
  4. The solution set is \(\emptyset\text{.}\)

Determine the domain of each of the following functions. State each domain using interval notation.

10

\(r(x)=4-\sqrt{8-16x}\)

Solution

The domain of \(r\) is \(\left(-\infty,\frac{1}{2}\right]\text{.}\)

11

\(w(t)=\sqrt[5]{6t+42}\)

Solution

The domain of \(w\) is \((-\infty,\infty)\text{.}\)

12

\(f(x)=2x^2-7x+8\)

Solution

The domain of \(f\) is \((-\infty,\infty)\text{.}\)

13

\(q(x)=\frac{y-9}{y+10}\)

Solution

The domain of \(q\) is \((-\infty,-10) \cup (-10,\infty)\text{.}\)

14

\(q(t)=\frac{\sqrt{t-12}}{t-3}\)

Solution

The domain of \(q\) is \([12,\infty)\text{.}\)

15

\(y(x)=\frac{x^2-5x+6}{x^2-8x+12}\)

Solution

The domain of \(y\) is \((-\infty,2) \cup (2,6) \cup (6,\infty)\text{.}\)

Determine the domain and range of functions presented in graphical form.

16

Determine the domain and range of the function \(g\) shown in FigureĀ 4.11.2. State the domain and range using interval notation.

plain text
Figure4.11.2\(y=g(x)\)

Solution

The domain of \(g\) is \((-\infty,1)\)and the range is \((-\infty,5]\text{.}\)

17

Determine the domain and range of the function \(z\) shown in FigureĀ 4.11.3. State the domain and range using interval notation.

plain text
Figure4.11.3\(y=z(x)\)

Solution

The domain of \(z\) is \((-5,0] \cup (3,6)\)and the range is \((-1,4]\text{.}\)

Evaluate and simplify each indicated expression.

18

Determine \(f(x+3)\) for the function \(f(x)=x^2-x\)

Solution

\(f(x+3)=x^2+5x+6\)

19

Determine \(g(-x)+5\) for the function \(g(x)=2x^3-3x^2-6x+2\text{.}\)

Solution

\(g(-x)+5=-2x^3-3x^2+6x+7\)

20

Determine \(3g(4t)\) for the function \(\sqrt{t-7}\text{.}\)

Solution

\(3g(4t)=3\sqrt{4t-7}\)

21

Determine \(-2y(t)-7\) for the function \(y(t)=-5t^2-11t+3\text{.}\)

Solution

\(-2y(t)-7=10t^2+22t-13\)

22

Determine \(3r(4t-3)\) for the function \(r(t)=\frac{3t-5}{t+4}\text{.}\)

Solution

\(3r(4t-3)=\frac{36t-42}{4t+1}\)

23

Determine \(\frac{w(y+5)-w(y-5)}{2}\) for the function \(w(y)=4y-7\text{.}\)

Solution

\(\frac{w(y+5)-w(y-5)}{2}=20\)

24

Evaluate and simplify \((f \circ f)(x)\) where \(f(x)=7-x\text{.}\)

Solution

\((f \circ f)(x)=x\)

25

Evaluate and simplify \((g \circ f)(x)\) where \(f(x)=3x+9\) and \(g(x)=5-x^2\text{.}\)

Solution

\((g \circ f)(x)=-9x^2-54x-76\)

26

Evaluate and simplify \((f \circ g)(4)\) where \(f(x)=3x+9\) and \(g(x)=5-x^2\text{.}\)

Solution

\((f \circ g)(4)=-24\)

27

Evaluate and simplify \((g \circ g)(8)\) where \(g(t)=\frac{3}{t+1}\text{.}\)

Solution

\((g \circ g)(8)=\frac{9}{4}\)

28

Evaluate and simplify \((k \circ h)(t)\) where \(h(t)=\frac{9t+2}{6}\) and \(k(t)=\frac{6t-2}{9}\text{.}\)

Solution

\(k \circ h)(t)=t\)

Determine the difference quotient for each of the following functions. Make sure that you completely simplify each expression.

29

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

Solution

The difference quotient is \(\frac{f(x+h)-f(x)}{h}=3,\,\,h \neq 0\text{.}\)

30

\(g(x)=x^2-5x+1\)

Solution

The difference quotient is \(\frac{g(x+h)-g(x)}{h}=2x+h-5,\,\, h \neq 0\text{.}\)

31

\(k(t)=3-2t^2\)

Solution

The difference quotient is \(\frac{k(t+h)-k(t)}{h}=-4t-2h,\,\, h \neq 0\text{.}\)

32

\(s(t)=-\frac{3}{2t}\)

Solution

The difference quotient is \(\frac{s(t+h)-s(t)}{h}=\frac{3}{2t^2+2th},\,\, h \neq 0\text{.}\)

Determine solution sets to inequalities based upon the graph of a function.

33

Determine the solution set to \(k(x) \geq 1\) based upon the function \(k\) shown in FigureĀ 4.11.4. State the solution set using both set-builder notation and interval notation.

plain text
Figure4.11.4\(y=k(x)\)

Solution

The solution set is \(\{x \mid -1 \leq x \lt 0 \text{ or } 2 \lt x \leq 5\}\text{.}\)

The solution set is \([-1,0) \cup (2,5]\text{.}\)

34

Determine the solution set to \(k(x) \leq 4\) based upon the function \(k\) shown in FigureĀ 4.11.5. State the solution set using both set-builder notation and interval notation.

plain text
Figure4.11.5\(y=k(x)\)

Solution

The solution set is \(\{x \mid x \leq 0 \text{ or } x \geq 4\}\text{.}\)

The solution set is \((-\infty,0] \cup [4,\infty)\text{.}\)

35

Determine the solution set to \(k(x) \gt 5\) based upon the function \(k\) shown in FigureĀ 4.11.6. State the solution set using both set-builder notation and interval notation.

plain text
Figure4.11.6\(y=k(x)\)

Solution

The solution set is \(\{\}\text{.}\)

The solution set is \(\emptyset\text{.}\)

36

Determine the solution set to \(k(x) \lt 6\) based upon the function \(k\) shown in FigureĀ 4.11.7. State the solution set using both set-builder notation and interval notation.

plain text
Figure4.11.7\(y=k(x)\)

Solution

The solution set is \(\{x \mid x \leq 0 \text{ or } x \gt 2\}\text{.}\)

The solution set is \((-\infty,0] \cup (2,\infty)\text{.}\)

Draw the inverse of each graphed function.

37
plain text
Figure4.11.8\(\highlightr{y=f(x)}\)
Solution
plain text
Figure4.11.9\(\highlightr{y=f(x)}\) and \(\highlight{y=f^{-1}(x)}\)
38
plain text
Figure4.11.10\(\highlightr{y=g(x)}\)
Solution
plain text
Figure4.11.11\(\highlightr{y=g(x)}\) and \(\highlight{y=g^{-1}(x)}\)

Given the formula for \(f(x)\text{,}\) determine the formula for \(f^{-1}(x)\text{.}\)

39

\(f(x)=4x-19\)

Solution

\(f^{-1}(x)=\frac{1}{4}x+\frac{19}{4}\)

40

\(f(x)=\frac{\sqrt[3]{2x+8}}{2}-6\)

Solution

\(f^{-1}(x)=\frac{1}{2}(2x+12)^3-4\)

41

\(f(x)=\frac{1-2x}{3x+11}\)

Solution

\(f^{-1}(x)=\frac{11-x}{3x+2}\)

The graph of a function named \(f\) is given in FigureĀ 4.11.12. Three functions are stated in terms of \(f\text{.}\) Graph each of these functions.

plain text
Figure4.11.12\(\highlight{y=f(x)}\)
42

\(g(x)=\frac{1}{3}f(3x-6)+2\)

Solution
plain text
Figure4.11.13\(\highlight{y=f(x)}\) and \(\highlightr{y=g(x)}\)
43

\(h(x)=\frac{2}{3}f\left(\frac{3}{2}(x+1)\right)-1\)

Solution
plain text
Figure4.11.14\(\highlight{y=f(x)}\) and \(\highlightr{y=h(x)}\)
44

\(k(x)=-f(-(x+2))\)

Solution
plain text
Figure4.11.15\(\highlight{y=f(x)}\) and \(\highlightr{y=k(x)}\)

One point on a function named \(f\) is \((12,-20)\text{.}\) Several functions are defined in terms of \(f\text{.}\) For each function, use graphical transformation properties to determine where the stated point on \(f\) resides on the graph of the new function.

45

\(g(x)=-f(4(x-3))-8\)

Solution

The point \((12,-20)\) for \(f\) moves to \((6,12)\) on \(g\)

46

\(h(x)=\frac{1}{10}f(\frac{1}{5}x+3)\)

Solution

The point \((12,-20)\) for \(f\) moves to \((45,-2)\) on \(h\)

47

\(k(x)=2f(-6x-12)+5\)

Solution

The point \((12,-20)\) for \(f\) moves to \((-4,-35)\) on \(k\)

48

\(w(x)=-\frac{2}{5}f(\frac{2}{3}(x+8))-3\)

Solution

The point \((12,-20)\) for \(f\) moves to \((10,5)\) on \(w\)