Question 1

Given $f(x)$ and $g(x)$, find the indicated composition and state its domain.
-$f(x)=8 x+7, g(x)=4 x-1$
Find $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})$.

Accepted Answer

A)  $32 x+15$, all real numbers 
B)  $32 x+6$, all real numbers 
C)  $32 x+27$, all real numbers 
D)  $32 x$ - 1, all real numbers 
A)  $32 x+15$, all real numbers 
B)  $32 x+6$, all real numbers 
C)  $32 x+27$, all real numbers 
D)  $32 x$ - 1, all real numbers

Question 2

Evaluate using the change-of-base formula. Round to four decimal places.
-$\log _{8} 0.50$

Accepted Answer

A)  -3.0000 
B)  -0.6666 
C)  -0.3333 
D)  0.3333 
A)  -3.0000 
B)  -0.6666 
C)  -0.3333 
D)  0.3333

Question 3

For the given functions $f(x)$ and $g(x)$, find a value $x$ for which $(f\circ g)(x)=(g\circ f)(x)$.
-$f(x)=x+6, g(x)=x^{2}-16$

Accepted Answer

A)  $x=-\frac{7}{2}$ 
B)  $x=\frac{13}{2}$ 
C)  $x=\frac{7}{2}$ 
D)  $x=-\frac{5}{2}$ 
A)  $x=-\frac{7}{2}$ 
B)  $x=\frac{13}{2}$ 
C)  $x=\frac{7}{2}$ 
D)  $x=-\frac{5}{2}$

Question 4

For the piecewise function $f(x)=\left\{\begin{array}{l}-|x+3|+14 \text { if } x \leq 7 \ \sqrt{x-5}-10 \text { if } x>7\end{array}\right.$, find $(f \circ f \circ f)(2)$.

Accepted Answer

A)  12 
B)  9 
C)  19 
D)  -8 
A)  12 
B)  9 
C)  19 
D)  -8

Question 5

For the given function $f(x)$, find $f^{-1}(x)$. State the domain of $f^{-1}(x)$.
-$f(x)=x^{2}-15(x \geq 0)$

Accepted Answer

A)  $f 1(x)=x^{2}+15, x \geq-15$ 
B)  $f^{-1}(x)=\sqrt{x-15}, x \geq 15$ 
C)  $f^{1}(x)=\sqrt{x^{2}+15}, x \geq-15$ 
D)  $f^{-1}(x)=\sqrt{x+15}, x \geq-15$ 
A)  $f 1(x)=x^{2}+15, x \geq-15$ 
B)  $f^{-1}(x)=\sqrt{x-15}, x \geq 15$ 
C)  $f^{1}(x)=\sqrt{x^{2}+15}, x \geq-15$ 
D)  $f^{-1}(x)=\sqrt{x+15}, x \geq-15$

Question 6

Rew rite as a single logarithm. A ssume all variables represent positive real numbers.
-$\log \sqrt[5]{\mathrm{x}}+\log \mathrm{x}^{3}-\log \mathrm{x}^{2}$

Accepted Answer

A)  $10 \log x$ 
B)  $\frac{4}{5} \log \mathrm{x}$ 
C)  $\frac{6}{5} \log \mathrm{x}$ 
D)  $6 \log \mathrm{x}$ 
A)  $10 \log x$ 
B)  $\frac{4}{5} \log \mathrm{x}$ 
C)  $\frac{6}{5} \log \mathrm{x}$ 
D)  $6 \log \mathrm{x}$

Question 7

Evaluate the given function.
-$f(x)=19 \log _{3}(x-3), f(12)$

Accepted Answer

A)  38 
B)  2 
C)  46.8344969 
D)  9.5 
A)  38 
B)  2 
C)  46.8344969 
D)  9.5

Question 8

Rew rite as a single logarithm. A ssume all variables represent positive real numbers.
-$6 \log _{2}(4 x+1)+2 \log _{2}(2 x-5)$

Accepted Answer

A)  $\log _{2}\left((4 x+1)^{6}+(2 x-5)^{2}\right)$ 
B)  $12 \log _{2}(4 x+1)(2 x-5)$ 
C)  $\log _{2}\left((4 x+1)^{6}(2 x-5)^{2}\right)$ 
D)  $\log _{2} \frac{(4 x+1)^{6}}{(2 x-5)^{2}}$ 
A)  $\log _{2}\left((4 x+1)^{6}+(2 x-5)^{2}\right)$ 
B)  $12 \log _{2}(4 x+1)(2 x-5)$ 
C)  $\log _{2}\left((4 x+1)^{6}(2 x-5)^{2}\right)$ 
D)  $\log _{2} \frac{(4 x+1)^{6}}{(2 x-5)^{2}}$

Question 9

Solve the problem.
-The number of male lawyers, in thousands, in a certain country in a particular year can be approximated by the function $\mathrm{f}(\mathrm{x})=11.1 \mathrm{x}+215$, where $\mathrm{x}$ represents the number of years after 1995. The number of female lawyers, in thousands, in a certain country in a particular year can be approximated by the function $\mathrm{g}(\mathrm{x})=10.1 \mathrm{x}+143$, where again $\mathrm{x}$ represents the number of years after 1995.
(i) Find $(f+g)(x)$. Explain, in your own words, what this function represents.
(ii) Find $(f+g)(45)$. Explain, in your own words, what this number represents.

Accepted Answer

A)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=154.1 \mathrm{x}+225.1$; this is the total number of lawyers, in thousands, in a certain country x years after 1995 .
(ii) 7159.6; there will be 7,159,600 lawyers in a certain country in the year 2040. 
B)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=379.2 \mathrm{x}$; this is the total number of lawyers, in thousands, in a certain country $\mathrm{x}$ years after 1995.
(ii) 17,064; there will be 17,064,000 lawyers in a certain country in the year 2040. 
C)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=21.2 \mathrm{x}+358$; this is the total number of lawyers, in thousands, in a œrtain country x years after 1995.
(ii) 1312; there will be 1,312,000 lawyers in a certain country in theyear 2040. 
D)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=226.1 \mathrm{x}+153.1$; this is the total number of lawyers, in thousands, in a certain country x years after 1995 .
(ii) 10,327.6; there will be 10,327,600 lawyers in a certain country in the year 2040. 
A)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=154.1 \mathrm{x}+225.1$; this is the total number of lawyers, in thousands, in a certain country x years after 1995 .
(ii) 7159.6; there will be 7,159,600 lawyers in a certain country in the year 2040. 
B)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=379.2 \mathrm{x}$; this is the total number of lawyers, in thousands, in a certain country $\mathrm{x}$ years after 1995.
(ii) 17,064; there will be 17,064,000 lawyers in a certain country in the year 2040. 
C)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=21.2 \mathrm{x}+358$; this is the total number of lawyers, in thousands, in a œrtain country x years after 1995.
(ii) 1312; there will be 1,312,000 lawyers in a certain country in theyear 2040. 
D)  (i) $(\mathrm{f}+\mathrm{g})(\mathrm{x})=226.1 \mathrm{x}+153.1$; this is the total number of lawyers, in thousands, in a certain country x years after 1995 .
(ii) 10,327.6; there will be 10,327,600 lawyers in a certain country in the year 2040.

Question 10

Graph. State the domain, range, and horizontal asymptote of the function
-$f(x)=e^{(x-2)}-2$

Accepted Answer

A)  Domain: $ (-\infty, \infty) $, range: $ (-2, \infty) $
Horizontal asymptote: $ \mathrm{y}=2 $
  
B)  Domain: $ (-\infty, \infty) $, range: $ (2, \infty) $
Horizontal asymptote: $ \mathrm{y}=-2 $ 
  
C)  Domain: $ (-\infty, \infty) $, range: $ (2, \infty) $
Horizontal asymptote: $ \mathrm{y}=2 $ 
  
D)  Domain: $ (-\infty, \infty) $, range: $ (-2, \infty) $ 
Horizontal asymptote: $ \mathrm{y}=-2 $
  
A)  Domain: $ (-\infty, \infty) $, range: $ (-2, \infty) $
Horizontal asymptote: $ \mathrm{y}=2 $
  
B)  Domain: $ (-\infty, \infty) $, range: $ (2, \infty) $
Horizontal asymptote: $ \mathrm{y}=-2 $ 
  
C)  Domain: $ (-\infty, \infty) $, range: $ (2, \infty) $
Horizontal asymptote: $ \mathrm{y}=2 $ 
  
D)  Domain: $ (-\infty, \infty) $, range: $ (-2, \infty) $ 
Horizontal asymptote: $ \mathrm{y}=-2 $

Question 11

For the given function $f(x)$, find $f^{-1}(x)$ and graph the function and its inverse.
-$ f(x)=\left(\frac{5}{2}\right)^{x} $

Accepted Answer

A)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
B)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
C)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
D)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
A)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
B)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
C)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $
  
D)  $ f(x)=\left(\frac{5}{2}\right)^{x}, f^{-1}(x)=\log _{5 / 2} x $

Question 12

The sales of a new product (in items per month) can be approximated by
$S(x)=375+400 \log (3 t+1)$, where $t$ represents the number of months after the item first becomes available. Find the number of items sold per month 3 months after the item first becomes available.

Accepted Answer

A)  4375 items per month 
B)  775 items per month 
C)  8375 items per month 
D)  1175 items per month 
A)  4375 items per month 
B)  775 items per month 
C)  8375 items per month 
D)  1175 items per month

Question 13

Solve.
-$2^{-4 x+12}=16$

Accepted Answer

A)  $\{8\}$ 
B)  $\{3\}$ 
C)  $\{2\}$ 
D)  $\{-2\}$ 
A)  $\{8\}$ 
B)  $\{3\}$ 
C)  $\{2\}$ 
D)  $\{-2\}$

Question 14

Determine the equation of the vertical asymptote for the graph of this function, and state the domain and range of this function.
-$f(x)=\ln (x+2)-4$

Accepted Answer

A)  Vertical asymptote: $\mathrm{x}=-2$;
Domain: $(-2, \infty)$, Range: $(-\infty, \infty)$ 
B)  Vertical asymptote: $x=-4$;
Domain: $(-4, \infty)$, Range: $(-\infty, \infty)$ 
C)  Vertical asymptote: $\mathrm{x}=4$;
Domain: $(4, \infty)$, Range: $(-\infty, \infty)$ 
D)  Vertical asymptote: $x=2$;
Domain: $(2, \infty)$, Range: $(-\infty, \infty)$ 
A)  Vertical asymptote: $\mathrm{x}=-2$;
Domain: $(-2, \infty)$, Range: $(-\infty, \infty)$ 
B)  Vertical asymptote: $x=-4$;
Domain: $(-4, \infty)$, Range: $(-\infty, \infty)$ 
C)  Vertical asymptote: $\mathrm{x}=4$;
Domain: $(4, \infty)$, Range: $(-\infty, \infty)$ 
D)  Vertical asymptote: $x=2$;
Domain: $(2, \infty)$, Range: $(-\infty, \infty)$

Question 15

Rewrite in exponential form.
-$\log x=4$

Accepted Answer

A)  $x^{10}=4$ 
B)  $10^{4}=x$ 
C)  $10^{x}=4$ 
D)  $4^{10}=\mathrm{x}$ 
A)  $x^{10}=4$ 
B)  $10^{4}=x$ 
C)  $10^{x}=4$ 
D)  $4^{10}=\mathrm{x}$

Question 16

Solve the problem.
-Find $\left(\frac{f}{g}\right)(4)$ when $f(x)=x-5$ and $g(x)=7 x+1$.

Accepted Answer

A)  $-\frac{1}{28}$ 
B)  $-\frac{3}{8}$ 
C)  $-\frac{1}{32}$ 
D)  $-\frac{1}{29}$ 
A)  $-\frac{1}{28}$ 
B)  $-\frac{3}{8}$ 
C)  $-\frac{1}{32}$ 
D)  $-\frac{1}{29}$

Question 17

Solve the equation.
-$\log _{4}(\mathrm{x}-1)+\log _{4}(\mathrm{x}-7)=2$

Accepted Answer

A)  $\varnothing$ 
B)  $\{-1\}$ 
C)  $\{9,-1\}$ 
D)  $\{9\}$ 
A)  $\varnothing$ 
B)  $\{-1\}$ 
C)  $\{9,-1\}$ 
D)  $\{9\}$

Question 18

Evaluate the given function. Round to the nearest thousandth.
-$f(x)=e^{x}, f(-4.67)$

Accepted Answer

A)  0.069 
B)  -0.931 
C)  -0.991 
D)  0.009 
A)  0.069 
B)  -0.931 
C)  -0.991 
D)  0.009

Question 19

Evaluate.
-$\log _{2} 8$

Accepted Answer

A)  -2 
B)  3 
C)  8 
D)  -8 
A)  -2 
B)  3 
C)  8 
D)  -8

Question 20

For the given functions f(x) and g(x), find (f + g)(x) or (f - g)(x) as indicated
-$f(x)=5 x-6, g(x)=2 x-4$
Find $(f-g)(x)$.

Accepted Answer

A)  $7 \mathrm{x}-10$ 
B)  $-3 x+2$ 
C)  $3 x-2$ 
D)  $3 x-10$ 
A)  $7 \mathrm{x}-10$ 
B)  $-3 x+2$ 
C)  $3 x-2$ 
D)  $3 x-10$

Given $f(x)$ and $g(x)$ , find the indicated composition and state its domain. - $f(x)=8 x+7, g(x)=4 x-1$ Find $(\mathrm{g} \circ \mathrm{f})(\mathrm{x})$ .

Evaluate using the change-of-base formula. Round to four decimal places. - $\log _{8} 0.50$

For the given functions $f(x)$ and $g(x)$ , find a value $x$ for which $(f\circ g)(x)=(g\circ f)(x)$ . - $f(x)=x+6, g(x)=x^{2}-16$

For the piecewise function $f(x)=\left\{\begin{array}{l}-|x+3|+14 \text { if } x \leq 7 \\ \sqrt{x-5}-10 \text { if } x>7\end{array}\right.$ , find $(f \circ f \circ f)(2)$ .

For the given function $f(x)$ , find $f^{-1}(x)$ . State the domain of $f^{-1}(x)$ . - $f(x)=x^{2}-15(x \geq 0)$

Rew rite as a single logarithm. A ssume all variables represent positive real numbers. - $\log \sqrt[5]{\mathrm{x}}+\log \mathrm{x}^{3}-\log \mathrm{x}^{2}$

Evaluate the given function. - $f(x)=19 \log _{3}(x-3), f(12)$

Rew rite as a single logarithm. A ssume all variables represent positive real numbers. - $6 \log _{2}(4 x+1)+2 \log _{2}(2 x-5)$

Graph. State the domain, range, and horizontal asymptote of the function - $f(x)=e^{(x-2)}-2$ $Graph. State the domain, range, and horizontal asymptote of the function - f(x)=e^{(x-2)}-2$

For the given function $f(x)$ , find $f^{-1}(x)$ and graph the function and its inverse. - $f(x)=\left(\frac{5}{2}\right)^{x}$ $For the given function f(x) , find f^{-1}(x) and graph the function and its inverse. - f(x)=\left(\frac{5}{2}\right)^{x}$

The sales of a new product (in items per month) can be approximated by $S(x)=375+400 \log (3 t+1)$ , where $t$ represents the number of months after the item first becomes available. Find the number of items sold per month 3 months after the item first becomes available.

Solve. - $2^{-4 x+12}=16$

Determine the equation of the vertical asymptote for the graph of this function, and state the domain and range of this function. - $f(x)=\ln (x+2)-4$

Rewrite in exponential form. - $\log x=4$

Solve the problem. -Find $\left(\frac{f}{g}\right)(4)$ when $f(x)=x-5$ and $g(x)=7 x+1$ .

Solve the equation. - $\log _{4}(\mathrm{x}-1)+\log _{4}(\mathrm{x}-7)=2$

Evaluate the given function. Round to the nearest thousandth. - $f(x)=e^{x}, f(-4.67)$

Evaluate. - $\log _{2} 8$

For the given functions f(x) and g(x), find (f + g)(x) or (f - g)(x) as indicated - $f(x)=5 x-6, g(x)=2 x-4$ Find $(f-g)(x)$ .

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Rational Expressions and Equations

Radical Expressions and Equations

Quadratic Equations

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 9: Logarithmic and Exponential Functions

Given f(x)f(x)f(x) and g(x)g(x)g(x) , find the indicated composition and state its domain. - f(x)=8x+7,g(x)=4x−1f(x)=8 x+7, g(x)=4 x-1f(x)=8x+7,g(x)=4x−1 Find (g∘f)(x)(\mathrm{g} \circ \mathrm{f})(\mathrm{x})(g∘f)(x) .

Evaluate using the change-of-base formula. Round to four decimal places. - log⁡80.50\log _{8} 0.50log8​0.50

For the given functions f(x)f(x)f(x) and g(x)g(x)g(x) , find a value xxx for which (f∘g)(x)=(g∘f)(x)(f\circ g)(x)=(g\circ f)(x)(f∘g)(x)=(g∘f)(x) . - f(x)=x+6,g(x)=x2−16f(x)=x+6, g(x)=x^{2}-16f(x)=x+6,g(x)=x2−16

For the piecewise function f(x)={−∣x+3∣+14 if x≤7x−5−10 if x>7f(x)=\left\{\begin{array}{l}-|x+3|+14 \text { if } x \leq 7 \\ \sqrt{x-5}-10 \text { if } x>7\end{array}\right.f(x)={−∣x+3∣+14 if x≤7x−5​−10 if x>7​ , find (f∘f∘f)(2)(f \circ f \circ f)(2)(f∘f∘f)(2) .

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)=x2−15(x≥0)f(x)=x^{2}-15(x \geq 0)f(x)=x2−15(x≥0)

Rew rite as a single logarithm. A ssume all variables represent positive real numbers. - log⁡x5+log⁡x3−log⁡x2\log \sqrt[5]{\mathrm{x}}+\log \mathrm{x}^{3}-\log \mathrm{x}^{2}log5x​+logx3−logx2

Evaluate the given function. - f(x)=19log⁡3(x−3),f(12)f(x)=19 \log _{3}(x-3), f(12)f(x)=19log3​(x−3),f(12)

Rew rite as a single logarithm. A ssume all variables represent positive real numbers. - 6log⁡2(4x+1)+2log⁡2(2x−5)6 \log _{2}(4 x+1)+2 \log _{2}(2 x-5)6log2​(4x+1)+2log2​(2x−5)

Graph. State the domain, range, and horizontal asymptote of the function - f(x)=e(x−2)−2f(x)=e^{(x-2)}-2f(x)=e(x−2)−2

For the given function f(x)f(x)f(x) , find f−1(x)f^{-1}(x)f−1(x) and graph the function and its inverse. - f(x)=(52)x f(x)=\left(\frac{5}{2}\right)^{x} f(x)=(25​)x

Solve. - 2−4x+12=162^{-4 x+12}=162−4x+12=16

Determine the equation of the vertical asymptote for the graph of this function, and state the domain and range of this function. - f(x)=ln⁡(x+2)−4f(x)=\ln (x+2)-4f(x)=ln(x+2)−4

Rewrite in exponential form. - log⁡x=4\log x=4logx=4

Solve the problem. -Find (fg)(4)\left(\frac{f}{g}\right)(4)(gf​)(4) when f(x)=x−5f(x)=x-5f(x)=x−5 and g(x)=7x+1g(x)=7 x+1g(x)=7x+1 .

Solve the equation. - log⁡4(x−1)+log⁡4(x−7)=2\log _{4}(\mathrm{x}-1)+\log _{4}(\mathrm{x}-7)=2log4​(x−1)+log4​(x−7)=2

Evaluate the given function. Round to the nearest thousandth. - f(x)=ex,f(−4.67)f(x)=e^{x}, f(-4.67)f(x)=ex,f(−4.67)

Evaluate. - log⁡28\log _{2} 8log2​8

For the given functions f(x) and g(x), find (f + g)(x) or (f - g)(x) as indicated - f(x)=5x−6,g(x)=2x−4f(x)=5 x-6, g(x)=2 x-4f(x)=5x−6,g(x)=2x−4 Find (f−g)(x)(f-g)(x)(f−g)(x) .