Preparing for Calculus

P.5 Assess Your Understanding
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Concepts and Vocabulary

The graph of every exponential function $f( x) =a^{x}$ , $a>0$ and $a\neq 1$ , passes through three points: _____, _____, and _____.

$\left(-1, \frac{1}{a}\right)$ , $(0, 1)$ , $(1, a)$

True or False The graph of the exponential function $f( x) =\left( \dfrac{3}{2}\!\right) ^{\!\!\!x}$ is decreasing.

False

If $3^{x}=3^{4}$ , then $x=$ _____.

If $4^{x}=8^{2}$ then $x=$ _____.

True or False The graphs of $y=3^{x}$ and $y=\left( \dfrac{1}{3}\!\right) ^{\!\!\!x}$ are symmetric with respect to the line $y=x$ .

False

True or False The range of the exponential function $f(x) = a^{x}$ , $a>0$ and $a\neq 1$ , is the set of all real numbers.

False

The number $e$ is defined as the base of the exponential function $f$ whose tangent line to the graph of $f$ at the point $( 0,1)$ has slope _____.

The domain of the logarithmic function $f( x) =\log _{a}x$ is _____.

$\{x | x > 0\}$

The graph of every logarithmic function $f( x) =\log _{a}x$ , $a>0$ and $a\neq 1$ , passes through three points: _____, _____, and _____.

$\left(\frac{1}{a}, -1\right)$ , $(1, 0)$ , $(a, 1)$

Multiple Choice The graph of $f( x) =\log _{2}x$ is [(a) increasing, (b) decreasing, (c) neither].

(a)

True or False If $y=\log _{a}x$ , then $y=a^{x}.$

False

True or False The graph of $f( x) =\log _{a}x$ , $a>0$ and $a\neq 1$ , has an $x$ -intercept equal to $1$ and no $y$ -intercept.

True

True or False $\ln e^{x}=x$ for all real numbers.

True

$\ln e$ = _____.

Explain what the number $e$ is.

Answers will vary.

What is the $x$ -intercept of the function $h( x) =\ln ( x+1) ?$

Practice Problems

Suppose that $g( x) =4^{x}+2$ .
1. What is $g( -1)$ ? What is the corresponding point on the graph of $g$ ?
2. If $g( x) =66$ , what is $x$ ? What is the corresponding point on the graph of $g$ ?

$g(-1)=\frac{9}{4}$ , $\left(-1, \frac{9}{4}\right)$
$x=3$ , $(3, 66)$

Suppose that $g( x) =5^{x}-3$ .
1. What is $g ( -1)$ ? What is the corresponding point on the graph of $g$ ?
2. If $g( x) =122$ , what is $x$ ? What is the corresponding point on the graph of $g$ ?

In Problems 19–24, the graph of an exponential function is given. Match each graph to one of the following functions:

$y=3^{-x}$
$y=-3^{x}$
$y=-3^{-x}$
$y=3^{x}-1$
$y=3^{x-1}$
$y=1-3^{x}$

(a)

(c)

(b)

In Problems 25–30, use transformations to graph each function. Find the domain and range.

$f( x) =2^{x+2}$

Domain: $(-\infty, \infty)$ , range: $(0, \infty)$

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

$f( x) =4\left( \dfrac{1}{3}\!\right) ^{\!\!\!x}$

Domain: $(-\infty, \infty)$ , range: $(0, \infty)$

$f( x) =\left( \dfrac{1}{2}\!\right) ^{\!\!\!-x}+1$

$f( x) =e^{-x}$

Domain: $(-\infty, \infty)$ , range: $(0, \infty)$

$f( x) =5-e^{x}$

In Problems 31–34, find the domain of each function.

$F( x) =\log _{2}x^{2}$

$\{x | x \neq 0\}$

$g( x) =8+5\ln ( 2x+3)$

$f( x) =\ln ( x-1)$

$\{x | x > 1\}$

$g( x) = \sqrt{\ln x}$

In Problems 35–40, the graph of a logarithmic function is given. Match each graph to one of the following functions:

$y=\log _{3}x$
$y=\log _{3}( -x)$
$y=-\log_{3}x$
$y=\log _{3}x-1$
$y=\log _{3}( x-1)$
$y=1-\log _{3}x$

(b)

(f)

(c)

In Problems 41–44, use the given function f to:

(a) Find the domain of f.
(b) Graph f.
(c) From the graph of f, determine the range of f.
(d) Find $f^{-1}$ , the inverse of f.
(e) Use $f^{-1}$ to find the range of f.
Graph $f^{-1}$ .

$f( x) =\ln ( x+4)$

${\rm a} \,\{x | x > -4\}$
$({\rm b \,{\rm and }\, \rm f})$
${\rm c} \,(-\infty, \infty)$
${\rm d} \,f^{-1}(x)=e^x -4$
${\rm e} \,(-\infty, \infty)$

$f( x) =\dfrac{1}{2}\log ( 2x)$

$f( x) =3e^{x}+2$

${\rm a} \,(-\infty, \infty)$
$({\rm b \,{\rm and }\, \rm f})$
${\rm c} \,\{y | y > 2\}$
${\rm d} \,f^{-1}(x)=\ln\left(\frac{x-2}{3}\right)$
${\rm e} \,\{y | y > 2\}$

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

How does the transformation $y=\ln ( x+c)$ , $c>0$ , affect the $x$ -intercept of the graph of the function $f( x) =\ln x?$

It shifts the $x$ -intercept $c$ units to the left.

How does the transformation $y=e^{cx}$ , $c>0$ , affect the $y$ -intercept of the graph of the function $f( x) =e^{x}?$

In Problems 47–62, solve each equation.

$3^{x^{2}}=9^{x}$

$x=0$ , $x=2$

$5^{x^{2}+8}=125^{2x}$

$e^{3x}=\dfrac{e^{2}}{e^{x}}$

$x=\frac{1}{2}$

$e^{4x}\cdot e^{x^{2}}=e^{12}$

$e^{1-2x}=4$

$x=\frac{1-\ln 4}{2}$

$e^{1-x}=5$

$5(2^{3x}) =9$

$x=\frac{\ln\left(\frac{9}{5}\right)}{3\ln 2}$

$0.3(4^{0.2x}) =0.2$

$3^{1-2x}=4^{x}$

$x=\frac{\ln 3}{\ln 36}$

$2^{x+1}=5^{1-2x}$

$\log _{2}(2x+1) =3$

$x=\frac{7}{2}$

$\log _{3}( 3x-2) =2$

$\log _{x}\left( \dfrac{1}{8}\!\right) =3$

$x=\frac{1}{2}$

$\log _{x}64=-3$

$\ln ( 2x+3) =2\ln 3$

$x=3$

$\dfrac{1}{2}\log_{3}x=2\log _{3}2$

In Problems 63–66, use graphing technology to solve each equation. Express your answer rounded to three decimal places.

$\log _{5}( x+1) -\log _{4}( x-2) =1$

$x\approx 2.787$

$\ln x=x$

$e^{x}+\ln x=4$

$x\approx 1.315$

$e^{x}=x^{2}$

1. If $f( x) =\ln ( x+4)$ and $g( x) =\ln ( 3x+1)$ , graph $f$ and $g$ on the same set of axes.
2. Find the point(s) of intersection of the graphs of $f$ and $g$ by solving $f( x) =g( x)$ . Label any intersection points on the graph drawn in (a).
3. Based on the graph, solve $f( x) > g( x).$

(a)
$\left(\frac{3}{2}, \ln\frac{11}{2}\right)$
$\left\{x | -\frac{1}{3} < x < \frac{3}{2}\right\}$

1. If $f( x) =3^{x+1}$ and $g( x) =2^{x+2}$ , graph $f$ and $g$ on the same set of axes.
2. Find the point(s) of intersection of the graphs of $f$ and $g$ by solving $f( x) =g( x) .$ Round answers to three decimal places. Label any intersection points on the graph drawn in (a).
3. Based on the graph, solve $f( x) >g( x).$

P.5 Assess Your UnderstandingPrinted Page 48

P.5 Assess Your Understanding
Printed Page 48