Deck 1: Functions and Limits

Question 1

If , find .

Answer 1

Question 2

Find the exact value of the expression.

Answer 2

Question 3

Find the exact value of the expression. $\log _{5} 20+\log _{5} 35-6 \log _{5} 2$

A) 4
B) 3
C) 1
D) 5
E) 2

Answer 3

1

Question 4

Simplify the expression. $e^{3 \ln 5}$

A) 8
B) 15
C) 125
D) 243

Question 5

Find the inverse function of $f(x)=\frac{x+1}{2 x+1}$ .

A) $f^{-1}(x)=-\frac{x-1}{2 x-1}$
B) $f^{-1}(x)=\frac{x}{2 x-1}$
C) $f^{-1}(x)=(x+1)(2 x+1)$
D) $f^{-1}(x)=-\frac{2 x+1}{x-1}$
E) $f^{-1}(x)=-(x+1)(2 x+1)$

Question 6

Find $f^{-1}$ (a) for the function f and the real number a. $f(x)=x^{3}+x-2$ ; $a=-2$

A) 2
B) -1
C) 1
D) 0

Question 7

Determine whether the function is one-to-one. $f(x)=\sqrt{1-x^{2}}$

A) No
B) Yes

Question 8

Find the exact value of the given expression. sin $\left(\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$

A) $-\frac{\sqrt{3}}{2}$
B) $-\frac{1}{2}$
C) 1
D) 0

Question 9

Use the laws of logarithms to expand the expression. ln $\left(\frac{x+5}{x-6}\right)^{\frac{1}{2}}$

A) $\frac{1}{2}$ ln $\left(x-\frac{5}{6}\right)$
B) $\frac{1}{2}$ ln [(x + 5)(x - 6)]
C) $\frac{1}{2}$ ln $\left(\frac{x+5}{x-6}\right)$
D) $\frac{1}{2}$ ln (x + 5) - $\frac{1}{2}$ ln (x - 6)

Question 10

Solve the equation. $2 e^{x+2}=5$

A) x = ln $\frac{5}{2}$ + 2
B) x = ln $\frac{2}{5}$ - 2
C) x = ln $\frac{2}{5}$ + 2
D) x = ln $\frac{5}{2}$ - 2

Question 11

Simplify the expression.

Question 12

When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by $Q(t)=Q_{0}\left(1-e^{-t / a}\right)$ (The maximum charge capacity $Q_{0}$ is and t is measured in seconds.) How long does it take to recharge the capacitor to 90% of capacity if $a=5$ ?

A) $-5 \ln (1)$ seconds
B) $-5 \ln \left(\frac{1}{10}\right)$ seconds
C) $-\ln \left(\frac{4}{9}\right)$ seconds
D) $-5 \ln (10)$ seconds
E) $-5 \ln \left(\frac{9}{10}\right)$ seconds

Question 13

Find a formula for the inverse of the function. $y=\ln (x+6)$

A) $y=e^{x}-6$
B) $y=e^{x}+6$
C) $y=-6 e^{x}$
D) $y=6 e^{x}$
E) $y=e^{x+6}$

Question 14

Suppose that the graph of $y=\log _{2} x$ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft? Rounded to the nearest mile.

A) 13015052.4 mi
B) 1084587.7 mi
C) 13,014,952 mi
D) 13,015,152 mi
E) 5.2 mi

Question 15

Solve each equation for x.
(a) (b)

Question 16

Find the exact value of the given expression. cos^-1 0

A) $\pi$
B) $\frac{\pi}{2}$
C) 2 $\pi$
D) $\frac{2}{\pi}$

Question 17

Find the inverse of F) Then sketch the graphs of f and $f^{-1}$ on the same set of axes. $f(x)=\sqrt{1-x^{2}}$ , $x \geq 0$

A) $f^{-1}(x)=-\sqrt{1+x^{2}}$ , $x \geq 0$
B) $f^{-1}(x)=-\sqrt{1-x^{2}}$ , $x \geq 0$
C) $f^{-1}(x)=\sqrt{1-x^{2}}$ , $x \geq 0$
D) $f^{-1}(x)=\sqrt{1+x^{2}}$ , $x \geq 0$

Question 18

Fill in the blanks.
Let , where . __________ __________

Question 19

If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is $n=f(t)=100\left(2^{\frac{t}{3}}\right)$ . When will the population reach 35,000? Round the answer to the nearest tenth.

A) 30.4 hours
B) 15.4 hours
C) 20.4 hours
D) 35.4 hours
E) 25.4 hours

Question 20

Use the laws of logarithms to write the expression as the logarithm of a single quantity. 4 ln 5 - $\frac{5}{6}$ ln (x + 6)

A) $\frac{5 \ln 625}{6(x+6)}$
B) $\ln 5^{4}-\ln (x+6)^{5 / 6}$
C) $\frac{4 \ln 5}{\frac{5}{6} \ln (x+6)}$
D) $\ln \frac{625}{(x+6)^{5 / 6}}$

Question 21

Find the exponential function $f(x)=C a^{x}$ whose graph is given.

A) $f(x)=2\left(\frac{1}{3}\right)^{x}$
B) $f(x)=3\left(\frac{1}{3}\right)^{x}$
C) $f(x)=\left(\frac{3}{2}\right)^{x}$
D) $f(x)=\frac{1}{3^{x}}$
E) $f(x)=\left(\frac{3}{4}\right)^{x}$

Question 22

Plot the graph of the function f in an appropriate viewing window. $f(x)=\frac{x^{4}}{x^{4}+3}$

A)
B)
C)
D)

Question 23

Find all solutions of the equation correct to two decimal places. $x^{3}-9 x^{2}-40=0$

A) $x=4.00$ , $x=9.05$
B) $x=9.44$
C) $x=-4.00$ , $x=9.05$
D) $x=0$ , $x=4.01$ , $x=9.05$
E) $x=0$ , $x=9.05$

Question 24

Plot the graph of the function f in an appropriate viewing window. $f(x)=x+0.03 \sin 50 x$

A)
B)
C)
D)

Question 25

Find all solutions of the equation correct to two decimal places. $\tan x=\sqrt{1-x^{2}}$

A) $-4.47$
B) 4.47
C) 0.65
D) $-1.04,1.04$
E) $-2.29,2.21$

Question 26

Use the Law of Exponents to rewrite and simplify the expression. $\frac{\sqrt{a \sqrt{b}}}{\sqrt[7]{a b}}$

A) $a^{\frac{5}{14}} b^{\frac{3}{28}}$
B) $\frac{a}{b^{\frac{3}{28}}}$
C) $\frac{a}{b}$
D) $\frac{a^{\frac{5}{14}}}{b^{\frac{3}{28}}}$
E) $\left(\frac{a}{b}\right)^{\frac{5}{14}}$

Question 27

Determine whether f is one-to-one.

Question 28

Find the domain, range, and x-intercept(s) of the function.

Question 29

Use the Law of Exponents to rewrite and simplify the expression. $\frac{x^{4 x} \cdot x^{5 x+1}}{x^{x-5}}$

A) $x^{8 x+6}$
B) $x^{8 x+7}$
C) $x^{6 x-10}$
D) $x^{3 x+9}$
E) $x^{4 \pi-8}$

Question 30

Starting with the graph of $y=e^{x}$ , find the equation of the graph that results from reflecting about the line $y=3$ .

A) $y=e^{x}$
B) $y=3$
C) $y=-e^{x}+6$
D) $y=e^{-x}+6$
E) $y=-e^{x+6}$

Question 31

Find the inverse of
f. Then sketch the graphs of f and on the same set of axes.
f(x) = ,

Question 32

Find the inverse of
f. Then use a graphing utility to plot the graphs of f and on the same set of axes.

Question 33

Starting with the graph of $y=e^{x}$ , write the equation of the graph that results from shifting 5 units right.

A) $y=e^{x+5}$
B) $y=e^{x}-5$
C) $y=e^{x-5}$
D) $y=e^{x}+5$
E) $y=5 e^{x-5}$

Question 34

Find the zero(s) of the function f to five decimal places. f (x) = $x^{3}$ - 5x + 1

A) -2.33006, 0.20164, 2.12842
B) -2.33006, 0.20164, 0.35414
C) -2.45756, 0.20164, 0.35414
D) -2.45756, 0.20164, 2.12842

Question 35

Find the points of intersection of the graphs of the functions. Express your answers accurate to five decimal places. f (x) = 0.5 $x^{3}$ - 1.7 $x^{2}$ + 2.3x - 2; $g(x)$ = 2.8x - 4.8

A) (-1.21519, -3.40253)
(1)46104, 8.83163)
(3)15415, 4.0909)
B) (-1.21519, -8.20253),
(3)15415, 4.03163)
C) (-1.21519, -8.20253),
(1)46104, -0.7091)
(3)15415, 4.03163)
D) (-1.21519, -3.40253)
(3)15415, 4.0909)

Question 36

The table gives the population of the United States, in millions, for the years 1900 - 2000. Use a graphing calculator with exponential regression capability to model the U.S. population since 1900. Use the model to estimate the population in 1965 and to predict the population in the year 2025.
$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \text { Year } & 1900 & 1910 & 1920 & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\\hline \begin{array}{c}\text { Population } \\\text { (millions) }\end{array} & 76 & 92 & 106 & 123 & 131 & 150 & 179 & 203 & 227 & 250 & 281 \\\hline\end{array}$

A) 186 million, 402 million
B) 191 million, 407 million
C) 196 million, 412 million
D) 181 million, 397 million
E) 176 million, 392 million

Question 37

The function $f(x)=\sqrt{5+c x^{2}}$ is graphed below.

A) $c=0$
B) $c>0$
C) $c<0$

Question 38

Under ideal conditions a certain bacteria population is known to double every three hours. Suppose that there are initially 45 bacteria. What is the size of the population after 15 hours?

A) 2880 bacteria
B) 180 bacteria
C) 1440 bacteria
D) 360 bacteria
E) 720 bacteria

Question 39

Find the range of the function. $y=5+\cos x$

A) $[4,6]$
B) $(-1,3)$
C) $(2, \infty)$
D) $(-\infty, \infty)$
E) $[-1,1]$

Question 40

The graph of f is given. Sketch the graph of on the same set of axes. 3

Question 41

Find the solutions of the equation in .

Question 42

Find all solutions of the equation correct to two decimal places.

Question 43

Plot the graph of the function in (a) the standard viewing window and (b) the indicated window.
f (x) = x , [-3, 3] [-5, 5]

Question 44

Use the table to evaluate the expression $(f \circ g)(6)$ . $X$

$\begin{array}{|c|c|c|c|c|c|c|}\hline &1 & 2 & 3 & 4 & 5 & 6 \\x \\\hline &3 & 2 & 1 & 0 & 1 & 2 \\f(x)\\\hline &6 & 5 & 2 & 3 & 4 & 6 \\g(x)\\\hline\end{array}$

A) 5
B) 2
C) 3
D) 4
E) 6

Question 45

Plot the graph of the function f in an appropriate viewing window.
f (x) = 5x - 3x + 2x + 33

Question 46

The graph of the function $f$ follows. Choose the graph of $y=f\left(\frac{x}{2}\right)+1$

A)
B)
C)
D)

Question 47

Express the function in the form of .

Question 48

What is $\sqrt[10]{x}$ , given that $H=f \circ g \circ h$ and $H(x)=\sqrt[10]{\sqrt{x}-3}$ ?

A) h(x)
B) g(x)
C) f(x)

Question 49

If $f(x)=x+5$ and $h(x)=4 x-10$ , find a function g such that $g \circ f=h$ .

A) $g(x)=4 x+30$
B) $g(x)=4 x$
C) $g(x)=x-30$
D)
E) $g(x)=x+30$

Question 50

Which of the following is the equation for the function $g(x)$ ?

A) $g(x)=-f(x)+6$
B) $g(x)=6 f(x)$
C) $g(x)=f(x)-6$
D) $g(x)=\frac{f(x)}{6}$
E) $g(x)=-f(x+6)$

Question 51

Which of the following graphs is the graph of the function? $f(x)=\sin |2 x|$

A) Graph 2
B) Graph 1
C) Graph 3

Question 52

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations. $y=4+2 x-x^{2}$

A)
B)
C)
D)

Question 53

Express the function in the form of . __________

Question 54

For what values of x is the inequality true? Write your answer correct to two decimal places.

Question 55

Let and . Find .

Question 56

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 45 cm/s. Express the radius r of this circle as a function of the time t (in seconds) and find if A is the area of this circle as a function of the radius.

Question 57

A spherical balloon with radius inches has volume .
Find a function that represents the amount of air required to inflate the balloon from a radius of inches to a radius of inches.

Question 58

Find the points of intersection of the graphs of the functions. Express your answers accurate to five decimal places. = 0.2 - 1.3x - 3.7; = -0.1 + 0.2x + 6.4

Question 59

Find if

Question 60

The graph of the function $f$ follows. Choose the graph of $y=f(|x|) .$

A)
B)
C)
D)

Question 61

A box with an open top is to be constructed from a rectangular piece of card board with dimensions $b=4$ in. by $a=28$ in. by cutting out equal squares of side $X$ at each corner and then folding up the sides as in the figure. Express the volume $V$ of the box as a function of $X$ .

A) $V(x)=x^{3}-64 x^{2}+112 x$
B) $V(x)=4 x^{3}+32 x^{2}+196 x$
C) $V(x)=4 x^{3}+64 x^{2}+112 x$
D) $V(x)=x^{3}-32 x^{2}+196 x$
E) $V(x)=4 x^{3}-64 x^{2}+112 x$

Question 62

A rectangle has perimeter 14 m. Express the area of the rectangle as a function $A(l)$ of the length $l$ of one of its sides.

A) $A(l)=7 l-l^{2}$
B) $A(l)=l-7 l^{2}$
C) $A(l)=14 l-l^{2}$
D) $A(l)=14 l+l^{2}$
E) $A(l)=7 l+l^{2}$

Question 63

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function. $F=\frac{9}{5} C+32$ What is the F-intercept and what does it represent?

A) $\frac{9}{5}$ , Fahrenheit temperature corresponding to $0^{\circ} \mathrm{C}$
B) $\frac{9}{5}$ , Celsius temperature corresponding to $32^{\circ} \mathrm{C}$
C) 32, Celsius temperature corresponding to $0^{\circ} F$
D) 0, Fahrenheit temperature corresponding to $32^{\circ} \mathrm{C}$
E) 32, Fahrenheit temperature corresponding to $0^{\circ} \mathrm{C}$

Question 64

Find the domain of the function $f(x)=\frac{x}{-2 \sin x-3}$

A) $[2,3]$
B) $[-3,-2]$
C) $\left[-\frac{3}{2}, \infty\right)$
D) $(-\infty, \infty)$

Question 65

The monthly cost of driving a car depends on the number of miles driven. Julia found that in October it cost her $\$ 200$ to drive 300 mi and in July it cost her $\$ 350$ to drive 600 mi. Express the monthly cost C as a function of the distance driven d assuming that a linear relationship gives a suitable model.

A) $C=-50 d+0.5$
B) $C=-50 d-0.5$
C) $C=0.5 d+50$
D) $C=2 d+50$
E) $C=0.5 d-50$

Question 66

Scientists have discovered that a linear relationship exists between the amount of flobberworm mucus secretions and the air temperature. When the temperature is 65°F, the flobberworms each secrete 16 grams of mucus a day; when the temperature is 95°F, they each secrete 22 grams of mucus a day. Find a function M(t) that gives the amount of mucus secreted on a given day, where t is the temperature of that day in degrees Fahrenheit.

A) $M(t)=0.2 t+16$
B) $M(t)=5 t+16$
C) $M(t)=5 t+3$
D) $M(t)=0.2 t+3$

Question 67

If $f(x)=4 x^{2}+2$ , find and simplify $\frac{f(1+h)-f(1)}{h}$ , where $h \neq 0$ .

A) $2+4 h^{2}$
B) $4+8 h$
C) $4 h$
D) $8+4 h$

Question 68

Find the domain of the function. $f(x)=\frac{7 x+4}{x^{3}}$

A) (-¥, 0) È (0, ¥)
B) $\left(-\infty,-\frac{4}{7}\right) \cup\left(-\frac{4}{7}, \infty\right)$
C) $\left(-\infty, \frac{4}{7}\right) \cup\left(\frac{4}{7}, \infty\right)$
D) (-¥, 0)

Question 69

Find the domain and sketch the graph of the function. What is its range? f (x) = $\left\{\begin{array}{ll}-x+2 & \text { if } x \geq 2 \\x^{2}-1 & \text { if } x<2\end{array}\right.$

A) D: (- $\infty$ , $\infty$ ); R: (- $\infty$ , 0] $\cup$ (3, $\infty$ )
B) D: (- $\infty$ , $\infty$ ); R: [0, $\infty$ )
C) D: (- $\infty$ , $\infty$ ); R: (- $\infty$ , 3]
D) D: (- $\infty$ , $\infty$ ); R: [-1, $\infty$ )

Question 70

The relationship between the Fahrenheit and Celsius temperature scales is given by the linear function. Complete the table and find the slope.

Question 71

Which of the following graphs is neither even nor odd?

A) $f(x)=\frac{4 x^{2}}{x^{4}+1}$
B) $f(x)=8 x^{3}+10 x^{2}+1$
C) $f(x)=x^{3}-9 x$

Question 72

Find an expression for the function $y=f(x)$ whose graph is the bottom half of the parabola $x+(6-y)^{2}=0$ .

A) $y=6-\sqrt{-x}$
B) $y=36-\sqrt{-x}$
C) $y=6+\sqrt{x}$
D) $y=36-x^{2}$
E) $y=6-x^{2}$

Question 73

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A measured in of the caves by the equation (a) The cave called mission impossible near puebla, mexico, has suface area of .
How many species of bats would expect to find in that cave?
(b) If you discover that species of bats live in cave estimate the area of the cave.

Question 74

If $f(x)=x^{2}-x+6$ , evaluate the difference quotient $\frac{f(a+h)-f(a)}{h}$ .

A) $2 a-h-6$
B) $2 a-6$
C) h
D) $2 a+h-6$
E) none of these

Question 75

The graphs of $f(x)$ and $g(x)$ are given. For what values of x is $f(x)=g(x)$ ?

A) 4, 2
B) 0
C) -1
D) $-2,5$
E) -4, 12

Question 76

If the point $(7,3)$ is on the graph of an even function, what other point must also be on the graph?

A) $(-7,-3)$
B) $(7,-3)$
C) $(0,0)$
D) $(-7,3)$
E) None of these

Question 77

Many physical quantities are connected by inverse square laws, that is, by power functions of the form $f(x)=k x^{-2}$ . In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move $\frac{1}{3} \text { th }$ the distance to the lamp. How much brighter is the light?

A) $\frac{4}{9}$ times
B) $\frac{9}{2}$ times
C) 9 times.
D) $\frac{3}{2}$ times
E) 9 times

Question 78

The graph of the function $f$ is given. State the value of $f(-0.4)$ .

A) $f(-0.4)=10$
B) $f(-0.4)=0$
C) $f(-0.4)=-10$
D) $f(-0.4)=8$
E) $f(-0.4)=-8$

Question 79

Determine whether the function whose graph is given is even, odd, or neither.

A) Even
B) Neither
C) Odd

Question 80

Sandy wishes to have a rectangular garden in her backyard. She has 40 ft of fencing with which to enclose her garden. Letting x denote the width of the garden, find a function f in the variable x that gives the area of the garden.

A) $f(x)=40 x-x^{2}, \quad 0$ $< x$ $<20$ .
B) $f(x)=20 x-x^{2}, \quad 0$ $< x$ $<20$
C) $f(x)=40 x-x^{2}, \quad 0$ $< x$ $<40$
D) $f(x)=20 x-x^{2}, \quad 0$ $< x$ $<20$ .