## Section 5.12 Additional Practice Related to Functions

Ā¶### Exercises Exercises

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

###### 1.

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

\(k(9)=-42\)

###### 2.

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

\(y(-3)=35\)

###### 3.

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

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

###### 4.

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

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

###### 5.

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

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

###### 6.

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

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

###### 7.

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

\(w(6)=3\)

###### 8.

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

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 5.12.1.

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

Determine the solution set to each of the following equations based upon the function \(k\) shown in FigureĀ 5.12.1.

- \(k(x)=1\)
- \(k(x)=4\)
- \(k(x)=5\)
- \(k(x)=6\)

- \(k(2)\) is undefined.
- \(k(0)=-3\)
- \(k(4)=4\)
- \(k(-3)=-5\)

- The solution set is \(\{-1,5\}\text{.}\)
- The solution set is \(\{4\}\text{.}\)
- The solution set is \(\{3\}\text{.}\)
- The solution set is \(\emptyset\text{.}\)

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

###### 10.

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

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

###### 11.

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

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

###### 12.

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

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

###### 13.

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

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

###### 14.

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

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

###### 15.

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

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 5.12.2. State the domain and range using interval notation.

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 5.12.3. State the domain and range using interval notation.

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 \(f(x)=x^2-x\)

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

###### 19.

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

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

###### 20.

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

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

###### 21.

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

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

###### 22.

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

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

###### 23.

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

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

###### 24.

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

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

###### 25.

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

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

###### 26.

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

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

###### 27.

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

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

###### 28.

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

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

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

###### 29.

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

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

###### 30.

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

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

###### 31.

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

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

###### 32.

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

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 5.12.4. State the solution set using both set-builder notation and interval notation.

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 5.12.5. State the solution set using both set-builder notation and interval notation.

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 5.12.6. State the solution set using both set-builder notation and interval notation.

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 5.12.7. State the solution set using both set-builder notation and interval notation.

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

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

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

The graph of a function named \(f\) is given in FigureĀ 5.12.12. Three functions are stated in terms of \(f\text{.}\) Graph each of these functions.

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.

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

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

###### 46.

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

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

###### 47.

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

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

###### 48.

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

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