Question 1

Graph f(x). State the domain, range, and horizontal asymptote of the function
-$f(x)=4^{x}-4$

Accepted Answer

A)  Domain: $ (-\infty, \infty) $, range: $ (0, \infty) $
Horizontal asymptote: $ \mathrm{y}=0 $ 
 
B)  Domain: $ (-\infty, \infty) $, range: $ (-4, \infty) $ 
Horizontal asymptote: $ \mathrm{y}=-4 $
 
C)  Domain: $ (-\infty, \infty) $, range: $ (4, \infty) $
 Horizontal asymptote: $ \mathrm{y}=4 $ 
 
D)  Domain: $ (-\infty, \infty) $, range: $ (0, \infty) $
Horizontal asymptote: $ \mathrm{y}=0 $

Question 2

For the given function $g(x)$, find a function $f(x)$ such that $(f \circ g)(x)=x$.
-$g(x)=x+5$

Accepted Answer

A)  $f(x)=5 x$ 
B)  $f(x)=x-5$ 
C)  $f(x)=x+5$ 
D)  $f(x)=\frac{x}{5}$

Question 3

Rewrite in exponential form.
-$\log _{3}(9 x-4)=1$

Accepted Answer

A)  $\left\{\frac{7}{5}\right\}$ 
B)  $\{2\}$ 
C)  No solution 
D)  $\left\{\frac{7}{9}\right\}$

Question 4

Determine whether the graph is the graph of a function.
-

Accepted Answer

A) True 
 B)False

Question 5

Determine whether the functions $f(x)$ and $g(x)$ are inverse functions.
-$f(x)=x+3, g(x)=x-3$

Accepted Answer

A) True 
 B)False

Question 6

For the given function $g(x)$, find a function $f(x)$ such that $(f \circ g)(x)=x$.
-$g(x)=4 x+7$

Accepted Answer

A)  $f(x)=\frac{7}{4} x$ 
B)  $f(x)=\frac{x-4}{7}$ 
C)  $f(x)=\frac{x-7}{4}$ 
D)  $f(x)=\frac{x+7}{4}$

Question 7

Rewrite as a single logarithm using the product rule for logarithms. A ssume all variables represent positive real numbers.
-$\log _{7} \mathrm{x}+\log _{7} \mathrm{y}$

Accepted Answer

A)  $\log _{14}(\mathrm{xy})$ 
B)  $\log _{14}(\mathrm{x}+\mathrm{y})$ 
C)  $\log _{7}(\mathrm{x}+\mathrm{y})$ 
D)  $\log _{7}(x y)$

Question 8

Solve the equation.
-$\log _{7}(6 x-1)+\log _{7} \mathrm{x}=1$

Accepted Answer

A)  $\left\{\frac{7}{6}\right\}$ 
B)  $\varnothing$ 
C)  $\left\{-1, \frac{7}{6}\right\}$ 
D)  $\{-1\}$

Question 9

A fixed point of a function $f(x)$ is a value for which $f(x)=x$. For the given function, find:
 a) Fixed point of $f(x)$
b) (f  $\circ f)(\mathbf{x})$
c) Fixed point of (f $\circ\mathbf{f})(\mathbf{x})$
-$f(x)=-\frac{1}{2} x+6$

Accepted Answer

A)  a) 2 b) $3 x-1$ c) 5 
B)  a) 8 b) $\frac{1}{4} x-3$ c) -1 
C)  a) 11 b) $3 x-1$ c) 11 
D)  a) 4 b) $\frac{1}{4} \mathrm{x}+3$ c) 4

Question 10

Graph. State the domain, range, and vertical asymptote of the function
-$ f(x)=\ln (x+3) $

Accepted Answer

A)  Domain: $ (3, \infty) $, range: $ (-3, \infty) $
Vertical asymptote: $ \mathrm{x}=3 $ 
 
B)  Domain: $ (-3, \infty) $, range: $ (-3, \infty) $ 
Vertical asymptote: $ x=-3 $
 
C)  Domain: $ (3, \infty) $, range: $ (-\infty, \infty) $ 
Vertical asymptote: $ \mathrm{x}=3 $
 
D)  Domain: $ (-3, \infty) $, range: $ (-\infty, \infty) $ 
Vertical asymptote: $ \mathrm{x}=-3 $

Question 11

Expand. A ssume that all variables represent positive real numbers.
-$\log _{7}\left(\frac{12 \sqrt{\mathrm{r}}}{\mathrm{s}}\right)$

Accepted Answer

A)  $\log _{7} 12 \cdot \frac{1}{2} \log _{7} \mathrm{~m} \div \log _{7} \mathrm{~s}$ 
B)  $\log _{7}(12 \sqrt{r})-\log _{7} s$ 
C)  $\log _{7} \mathrm{~s}-\log _{7} 12-\frac{1}{2} \log _{7} \mathrm{r}$ 
D)  $\log _{7} 12+\frac{1}{2} \log _{7} r-\log _{7} s$

Question 12

The number of reports of a certain virus has increased exponentially since 1960. The number of cases can be approximated using the function $r(t)=207 \mathrm{e}^{0.006 t}$, where $t$ is the number of years since 1960. Estimate the number of cases in the year 2000.

Accepted Answer

A)  248 cases 
B)  197 cases 
C)  263 cases 
D)  526 cases

Question 13

Solve.
-$4^{x}=\frac{1}{64}$

Accepted Answer

A)  $\left\{\frac{1}{16}\right\}$ 
B)  $\{3\}$ 
C)  $\{-3\}$ 
D)  $\left\{\frac{1}{3}\right\}$

Question 14

For the given exponential function, find a) the average rate of change for the interval $[3,5]$ and b) the interval $[3.5,4.5]$, as well as c) the rate of change at $x=4$. Round all calculations to the nearest hundredth.
-$\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}-3}-5$

Accepted Answer

A)  a) 2.72 b) 2.83 c) 3.19 
B)  a) 3.19 b) 2.83 c) 2.72 
C)  a) 2.72 b) 2.96 c) 3.58 
D)  a) 3.58 b) 2.96 c) 2.72

Question 15

For the given function $f(x)$, find $f^{-1}(x)$. State the domain of $f^{-1}(x)$.
-$f(x)=\sqrt{4 x+5}$

Accepted Answer

A)  $f^{-1}(x)=\frac{2 x-5}{4}, x \geq 0$ 
B)  $f^{-1}(x)=\frac{x^{2}}{4}-5, x \geq 0$ 
C)  $\mathrm{f}^{-1}(\mathrm{x})=\frac{(\mathrm{x}-5)^{2}}{4}, \mathrm{x} \geq 0$ 
D)  $f^{-1}(x)=\frac{x^{2}-5}{4}, x \geq 0$

Question 16

Evaluate. Round to the nearest thousandth, if necessary.
-$\log 285$

Accepted Answer

A)  2.456 
B)  5.652 
C)  2.453 
D)  2.455

Question 17

Evaluate using the change-of-base formula. Round to four decimal places.
-$\log _{8.5} 137$

Accepted Answer

A)  0.4350 
B)  2.2990 
C)  16.1176 
D)  2.1367

Question 18

Graph f(x). State the domain, range, and horizontal asymptote of the function
-$f(x)=3^{x+3}$

Accepted Answer

A)  Domain: $ (-\infty, \infty) $, range: $ (-3, \infty) $
Horizontal asymptote: $ \mathrm{y}=0 $
 
B)  Domain: $ (-\infty, \infty) $, range: $ (0, \infty) $
Horizontal asymptote: $ \mathrm{y}=-3 $
 
C)  Domain: $ (-\infty, \infty) $, range: $ (3, \infty) $
Horizontal asymptote: $ y=3 $
 
D)  Domain: $ (-\infty, \infty) $, range: $ (0, \infty) $
Horizontal asymptote: $ y=0 $

Question 19

Find the specified domain
-For $g(x)=\sqrt{x+3}$ and $h(x)=\frac{1}{x-4}$, what is the domain of $h \circ g$ ?

Accepted Answer

A)  $[-3, \infty)$ 
B)  $[-3,13) \cup(13, \infty)$ 
C)  $[0,13) \cup(13, \infty)$ 
D)  $[-3,4) \cup(4, \infty)$

Question 20

Solve.
-$3^{1+2 \mathrm{x}}=243$

Accepted Answer

A)  $\{-2\}$ 
B)  $\{6\}$ 
C)  $\{2\}$. 
D)  $\{81\}$

Exam 9: Logarithmic and Exponential Functions

For the given function $g(x)$ , find a function $f(x)$ such that $(f \circ g) (x) =x$ . - $g(x) =x+5$

Rewrite in exponential form. - $\log _{3}(9 x-4) =1$

Determine whether the graph is the graph of a function. -

Determine whether the functions $f(x)$ and $g(x)$ are inverse functions. - $f(x)=x+3, g(x)=x-3$

For the given function $g(x)$ , find a function $f(x)$ such that $(f \circ g) (x) =x$ . - $g(x) =4 x+7$

Rewrite as a single logarithm using the product rule for logarithms. A ssume all variables represent positive real numbers. - $\log _{7} \mathrm{x}+\log _{7} \mathrm{y}$

Solve the equation. - $\log _{7}(6 x-1) +\log _{7} \mathrm{x}=1$

A fixed point of a function $f(x)$ is a value for which $f(x) =x$ . For the given function, find: a) Fixed point of $f(x)$ b) (f $\circ f) (\mathbf{x})$ c) Fixed point of (f $\circ\mathbf{f}) (\mathbf{x})$ - $f(x) =-\frac{1}{2} x+6$

Expand. A ssume that all variables represent positive real numbers. - $\log _{7}\left(\frac{12 \sqrt{\mathrm{r}}}{\mathrm{s}}\right)$

The number of reports of a certain virus has increased exponentially since 1960. The number of cases can be approximated using the function $r(t) =207 \mathrm{e}^{0.006 t}$ , where $t$ is the number of years since 1960. Estimate the number of cases in the year 2000.

Solve. - $4^{x}=\frac{1}{64}$

For the given exponential function, find a) the average rate of change for the interval $[3,5]$ and b) the interval $[3.5,4.5]$ , as well as c) the rate of change at $x=4$ . Round all calculations to the nearest hundredth. - $\mathrm{f}(\mathrm{x}) =\mathrm{e}^{\mathrm{x}-3}-5$

For the given function $f(x)$ , find $f^{-1}(x)$ . State the domain of $f^{-1}(x)$ . - $f(x) =\sqrt{4 x+5}$

Evaluate. Round to the nearest thousandth, if necessary. - $\log 285$

Evaluate using the change-of-base formula. Round to four decimal places. - $\log _{8.5} 137$

Find the specified domain -For $g(x) =\sqrt{x+3}$ and $h(x) =\frac{1}{x-4}$ , what is the domain of $h \circ g$ ?

Solve. - $3^{1+2 \mathrm{x}}=243$

Exam 9: Logarithmic and Exponential Functions

Graph f(x) . State the domain, range, and horizontal asymptote of the function - f(x) =4x−4f(x) =4^{x}-4f(x) =4x−4

For the given function g(x) g(x) g(x) , find a function f(x) f(x) f(x) such that (f∘g) (x) =x(f \circ g) (x) =x(f∘g) (x) =x . - g(x) =x+5g(x) =x+5g(x) =x+5

Rewrite in exponential form. - log⁡3(9x−4) =1\log _{3}(9 x-4) =1log3​(9x−4) =1

Determine whether the graph is the graph of a function. -

Determine whether the functions f(x)f(x)f(x) and g(x)g(x)g(x) are inverse functions. - f(x)=x+3,g(x)=x−3f(x)=x+3, g(x)=x-3f(x)=x+3,g(x)=x−3

For the given function g(x) g(x) g(x) , find a function f(x) f(x) f(x) such that (f∘g) (x) =x(f \circ g) (x) =x(f∘g) (x) =x . - g(x) =4x+7g(x) =4 x+7g(x) =4x+7

Rewrite as a single logarithm using the product rule for logarithms. A ssume all variables represent positive real numbers. - log⁡7x+log⁡7y\log _{7} \mathrm{x}+\log _{7} \mathrm{y}log7​x+log7​y

Solve the equation. - log⁡7(6x−1) +log⁡7x=1\log _{7}(6 x-1) +\log _{7} \mathrm{x}=1log7​(6x−1) +log7​x=1

Graph. State the domain, range, and vertical asymptote of the function - f(x) =ln⁡(x+3) f(x) =\ln (x+3) f(x) =ln(x+3)

Expand. A ssume that all variables represent positive real numbers. - log⁡7(12rs) \log _{7}\left(\frac{12 \sqrt{\mathrm{r}}}{\mathrm{s}}\right) log7​(s12r​​)

The number of reports of a certain virus has increased exponentially since 1960. The number of cases can be approximated using the function r(t) =207e0.006tr(t) =207 \mathrm{e}^{0.006 t}r(t) =207e0.006t , where ttt is the number of years since 1960. Estimate the number of cases in the year 2000.

Solve. - 4x=1644^{x}=\frac{1}{64}4x=641​

For the given function f(x) f(x) f(x) , find f−1(x) f^{-1}(x) f−1(x) . State the domain of f−1(x) f^{-1}(x) f−1(x) . - f(x) =4x+5f(x) =\sqrt{4 x+5}f(x) =4x+5​

Evaluate. Round to the nearest thousandth, if necessary. - log⁡285\log 285log285

Evaluate using the change-of-base formula. Round to four decimal places. - log⁡8.5137\log _{8.5} 137log8.5​137

Graph f(x) . State the domain, range, and horizontal asymptote of the function - f(x) =3x+3f(x) =3^{x+3}f(x) =3x+3

Find the specified domain -For g(x) =x+3g(x) =\sqrt{x+3}g(x) =x+3​ and h(x) =1x−4h(x) =\frac{1}{x-4}h(x) =x−41​ , what is the domain of h∘gh \circ gh∘g ?

Solve. - 31+2x=2433^{1+2 \mathrm{x}}=24331+2x=243