Section 1.1 – What is a Function? P. 5 #2,4,5,7,10,11,13,15,16,17,20,26 – Calculator Required
2. Each function f shown graphically below has the interval [-6, 6] as its domain. Find the range, zeros and f(-2) for each
function.
4. The function f shown graphically below has the interval [-5, 5] as its domain.
a.
b.
c.
d.
e.
How many zeroes does this function have?
Give approximate values for f(-3) and f(2).
Is the function increasing or decreasing near x = -1?
Is the graph concave up or concave down near x = -2?
List all intervals on which the function is increasing.
5. The function f shown graphically below has the interval [-5, 5] as its domain.
a.
b.
c.
d.
e.
What is the range of f?
On which interval(s) is f increasing?
Approximate f(1) and f(4).
On which intervals is f concave up?
Is the function increasing or decreasing at x = 0?
For problems 7, 10, 11, 13:
a. State the domain.
7. f ( x )=
11. f ( x ) =
x +1− 2
( x + 1)( x + 2 )
b. State the range
c. Find the zeros of the function.
10. f ( x ) =
( x − 2) 3 − x
13. f ( x ) =
1
x −9
2
15. Sketch a smooth, continuous curve which passes through the point P(2, 3) and which satisfies each of the following
conditions:
a. Concave up to the left of P
b. Concave down to the right of P
c. Increasing for x > 0
d. Decreasing for x < 0
16. Sketch a smooth, continuous curve which passes through the point P(0, 3) and which satisfies each of the following
conditions:
a. Decreasing on (-2, 3)
b. Concave down to the left of P
c. Concave up to the right of P
d. Increasing on ( 3, ∞ )
17. An open box is made by cutting squares of side x from the four corners of a sheet of cardboard that is 8.5 inches by
11 inches and then folding up the sides.
a. Express the volume of the box as a function of x.
b. Estimate the value of x that maximizes the volume of the box.
20. An 8.5 inch by 11 inch piece of paper contains a picture with a uniform border. The distance from the edge of the
paper to the picture is x inches on all sides.
a. Express the area of the picture as a function of x.
b. What are the domain and range of the function.
26. Express the surface area of a cube as a function of its volume.
Section 1.2 – Basic Functions and Transformations - P. 12 #13,15,17,19,22,23,25 – Calculator Required
In Exercises 13-19, the order in which transformations are to be applied to the graph of a given function is specified.
Given an equation for the transformed function.
1
15. y = ; shift right 1, shift up 3
13. y = x ; vertical stretch by 3, shift up 4.
x
19. y = x ; vertical stretch by 2, shift down 3
17. y = x ; shift left 2, shift up 3
22. The function f is defined by f ( x=
) x 2 − x . Graph each of the following functions noting its relationship to the original
function.
a. y = f ( x )
b. y = f ( x )
c. y =
1
f (x)
1
. Use the results of the previous exercise to predict the appearance of each of
x
the graphs of the following functions, then check by graphing each function.
1
b. y = g ( x )
c. y =
a. y = g ( x )
g(x)
23. The function g is defined by g ( x ) =
25. The graph of a function g with domain [-2, 4] is shown in the figure below. Sketch a graph of the following functions
and specify the domain and range.
y g(x) − 1
a.=
y g ( x − 1)
b.=
c. y = −g ( x )
y g ( x + 2)
d.=
−g ( x ) + 2
e. y =
f. y = 2g ( x )
g. y = 0.5g ( x )
h. y= g ( x + 1) − 2
Section 1.3 – Linear Functions and Mathematical Modeling – P. 22 #2,3,4,15ac – Calculator Required
2. Find an equation of the line in point-slope form that passes through the points (6, 1) and (-3, 2).
3. Find the equation of the line in point-slope form that has intercepts at (0, 4) and (-3, 0).
4. Find the equation of the line in point-slope form through the point (2, 1) that is
a. perpendicular to the line y = 3x – 5.
b. parallel to the line y = 2x + 3.
15. You need a Lear jet for 1 day. Knowing that Swissair rents a Lear jet with pilot for $2000 a day and $1.75 per mile,
while Air France rents a Lear jet for $1500 a day and $2.00 per mile, find the following:
a. For each company, write a formula giving cost as a function of distance travelled.
c. If cost were the only issue, when would you rent from Air France?
Section 1.4 – Exponential Functions – P. 30 #1,3,4,5,11,12,13 – Calculator Required
1. Given below is a list of six exponential functions, each with a rule of the form y = b x .
x
x
x
x
x
=
y1 2=
y 2 0.5=
y3 1.2=
y 4 0.2=
y5 0.8=
y 6 3.5 x
To help answer the following questions, display a graph and label values for each function.
a. Which of the functions are increasing? Decreasing?
b. Among those increasing, which has the greatest rate of increase? Which has the least?
c. Among those decreasing, which has the smallest rate of decrease?
d. Is there any input for which two of the six functions have the same output?
e. For any exponential function f ( x ) = b x , describe how the base b affects the shape of the curve.
3. Describe how the graph of each of the following functions results from transformation on the graph of y = 2x .
a. y = 2x −1
( )
d. y = − 2− x
 1
b. y =  
2
x
(
c. y = 2 2x +1
e.=
y 2x −1 − 1
)
f. y = 2x/2
4. Use your calculator to graph each of the following functions. Use the graph to determine the domain, the range, and
the zeros of each function.
a.=
y 2 x −3 − 1
(
)
=
b. y 2 3 x + 2 + 3
5. Find a rule for the function that results from each of the following transformations on the graph of y = 3 x .
a. Shift up 3 and then reflect about the x-axis.
b. Reflect about the y-axis.
c. Reflect about the y-axis, then reflect about the x-axis.
d. Shift right 2, then shift up 3.
11. The new Phillips sports car sells for $14,000 new. Because it is so well built it only depreciates at 12% per year.
a. If the Phillips depreciates linearly at 12% of the original value each year, how much will it be worth in five years?
b. If the Phillips depreciates exponentially at 12% each year, how much will it be worth in five years?
12. The population of mosquitoes around Rabbit Pond grows exponentially in the early summer. The population was
measured to be 1000 mosquitoes per 100 sq feet on May 23. The mosquito population doubles every 6 days during
this time of the year around Rabbit Pond.
a. Write an exponential function that relates the mosquito population per 100 sq feet to the time, t days, after May 23.
b. Determine the mosquito population on June 4.
c. How long does it take for the mosquito population to reach 5000 per 100 sq feet?
13. George has $7,000 to invest in a money market fund. He decides that his best choice is to invest in a fund that pays
5.75% interest compounded annually.
a. Write a function that relates the amount of money George has in his account to the number of years it has been
invested.
b. How much money will George have after 3 years?
‘
c. How long will it take George to double his money?
Section 1.5 – The Number e – P. 37 #1,3,5,7,9,11 – Calculator Required
For problems 1, 3, 5, convert the following functions from the form f ( x ) = A 0 erx to the form f ( x ) = A 0 b x . (Hint: er = b .)
Indicate which functions are increasing which are decreasing.
1. f ( x ) = 13e0.2x
3. P ( t ) =
e−0.67t
2
5. h ( x ) = 1705e −2x
For problems 7, 9, 11, convert the following functions from f ( x ) = A 0 b x to f ( x ) = A 0 erx . (Hint: Graphically solve the
equation er = b for r.)
( )
7. A ( x ) = A 0 2x
9. P ( x ) = 16.5 (1.2 )
x
11. k ( x ) =
275
3x
Section 1.6 – Inverse Functions – P. 42 # 1,5,7,9,12,13,14,15,17,19,21,23,28,31 – Calculator Required
1. A function is y = f(x) is defined by the table below. List the inverse of f, f −1 , and state the domain and range of f −1 .
x
-1
1
3
5
7
y
0
2
10
26
50
For problems 5 and 7, sketch the given function and its inverse on the same coordinate system. (Hint: Use the fact that if
(a, b) ∈ f , then (b, a) ∈ f −1 .) State the domain and range of f and f −1 .
5.
7.
9. Use a calculator and horizontal line test to determine whether or not the given function f is one-to-one.
a. f ( x=
) x3 + x
b. f ( x ) = x3 − 4x 2 + x − 10
c. f ( x ) = 0.1x3 − 0.1x 2 − 0.005x + 1
d. f ( x ) = x5 + 2x 4 − x 2 + 4x − 5
For problems 12-15, determine which of the following functions has an inverse. If the function has an inverse, sketch the
graph of the inverse. If the function does not have an inverse, explain how to limit the domain of the function so that it will
have an inverse.
In working with pairs of functions which are inverses of each other, it is often useful to think of one function, f, performing a
series of operations on a given input, while the inverse function, f −1 , undoes whatever the function f does. Use this
concept to determine the inverses of the following functions. Specify the domain and range of both the function and its
inverse.
17. F ( x ) =
3
x
19. H ( =
x)
23. J ( x ) =
21. g ( x ) =x 2 − 2 ( x ≥ 0 )
x −1
1
x
28. If f is an increasing (or decreasing) function, then f is one-to-one. Why?
31. Let f ( x )= 1 − x3 .
a. Find f −1 ( x ) .
b. How many solutions are there to the equation f ( x ) = f −1 ( x ) ?
Section 1.7 – Logarithms – P. 49 #3,4,5,6,7,9,10,11,12,29 – Calculator Required
3. Use the properties of logarithms to express the following as an algebraic expression involving log x, log y, and/or log z.
x 2 y3
xy
z
b. log
c. log x 4 y z
d. log
a. log
z
z
xy
4. Express each of the following as the logarithm of a single expression.
a. 2 log x + 4 log y − log13
b. log ( x + 1) − 2 log x − log y
log x
d. 2 log6 2 + log6 3 + log6 18
2
For problems 5, 6, 7, 9, 10, 11, 12 use logarithms to solve:
6. 2x + 5 =
7. 7.01
9. 5230e x = 14756
5. 5 x = 7
17
= 32x − 2.4
c. log 7 + 5 log y −
10. 5e x +1 = 27
11. 3 ⋅ 22x +1 =
24
12. e x
29. Solve for x:
a. log3 ( x − 4 ) ≤ 2
2
+ 2x
=2
3
b. log2 ( 3x − 2 ) − log2 ( +1) =
3
c. log2 ( 7 − x ) − log2 ( 5 − x ) =
d.
0
(log2 x )2 − 3 log2 x − 4 =
Section 1.8 – Combining Functions; Polynomial and Rational Functions - P. 58 #1,3*,11,12,19*,20 – Calculator Required
1. A table of values for functions f and g is given below.
x
-1
0
1
f(x)
3
4
-2
g(x)
3
1
-7
Using the tables, find
a. ( f + g)( 4 )
b. ( g − f )( −1)
f
d.   ( 0 )
 g
2
6
0
3
2
-1
4
-1
2
c.
( f ⋅ g)(1)
3
e. all x such that ( f − g)( x ) =
3. Determine from the graph whether the function is odd, even, or neither.
2x
b. f ( x ) =
c. f ( x )= x − x3
a. f =
( x ) 2x 2 − 1
2
x −1
x
x2
d. f ( x ) = x 4 − x 2 + 1
e. f ( x ) =
f. f ( x ) =
3
3
x +1
x −1
11. Determine a rational function that has zeros at -2 and 3, vertical asymptotes at x = 2 and x = -1, and a horizontal
asymptote at y = 1.
12. Find a rational function with vertical asymptotes at x = -1 and x = 2 and with the line y = 3 as a horizontal asymptote.
2
19. Consider the functions
f ( x ) x=
and g ( x ) 2x .
=
a.
b.
c.
d.
How many positive solutions are there to the equation f(x) = g(x)?
In the viewing rectangle 0 < x < 3 and 0 < y < 10, which function is growing faster?
In the viewing rectangle 0 < x < 8 and 0 < y < 100, which function is growing faster?
What are the positive solution(s) to f(x) = g(x)?
4
20. Consider the functions
=
f ( x ) x=
and g ( x ) 3 x .
a.
b.
c.
d.
e.
In the viewing rectangle 0 < x < 5, which function grows faster?
In the viewing rectangle 0 < x < 10, which function grows faster?
Find the positive solution(s) to f(x) = g(x). Round your answer to three decimal places.
When x = 3, which function has the larger value?
When x = 10, which function has the larger value?
Section 1.9 – Composition of Functions - P. 66 #1,2,3abc, 5abc,10,12abcd – Calculator Required
1. Each function below has the interval [-5, 5] as its domain.
Find:
a. [ f  g] ( −2 )
d.
b.
[f  f ] (5)
e.
2. Given tables for functions f and g
x
f(x)
g(x)
Find
a. [ f  g] ( 3 )
d.
[g  g] ( 4 )
-1
2
3
[ f  g] ( 2 )
[g  g] ( −2 )
0
4
4
b.
c.
[g  f ] ( −1)
f. all inputs x for which g g ( x )  = −1
1
3
2
2
0
6
3
1
2
[g  f ] ( 2)
4
-1
-1
c.
e. all inputs x for which [ f  g] ( x ) = 2
3. Tables of values for functions f and g are given below
x
-2
0
2
4
f(x)
0
1
-1
2
Find
a. [ f  f ] ( −2 )
b. [ f  g] ( 0 )
x
g(x)
-1
-2
[f  f ] ( 4)
0
2
c.
1
3
2
1
[g  f ] (1)
5. Let f ( x ) =
x3 , g ( x ) =
5x + 1, and h ( x ) =
2x . Find a formula for each function and specify its domain.
a. f ( g ( x ) )
b. h ( f ( x ) )
(
c. h ( g ( x ) )
)
ln x 2 + 1 and f ( x ) =
ln x, what is g(x)?
10. a. If [ f  g] ( x ) =
e x +1 − 7 and f ( x ) =
x − 7, determine g(x) and h(x).
b. If [ f  g  h] ( x ) =
2
12. If f(x) = x – 2 and g(x) = x + 3, h(x) = ln x, create compositions of f, g, and h to shift the graph of h as specified below.
a. Shift left 3
b. Shift down 2
c. Reflect through x-axis then shift up 3
d. Shift right 2 and up 3
Section 1.10 – Trigonometric Functions - P. 73 #4,9,10,12 – Calculator Required
4. State the domain, range and period of the following functions.
π

=
b. y = − cos ( 2x − π )
c. y = −2 sin ( πx + π )
a. y 2 sin  x − 
4


9. Given the function f ( x ) = x sin x .
π

=
d. y 6 cos  3x + 
2


a. Determine the zeros of f in the interval [0, 2π] .
b. For what arguments a does f(a) = a?
10. A ball is bouncing up and down. Its height (fee) above the ground at time x (sec) is given
by y
=
3 cos ( 4πx ) . How
many times does the ball hit the ground from x = 0 to x = 4?
12. On January 1, 1992, high tide in Seattle was at midnight. The water level at high tide was 9.9 feet and, at time t
πt
hours, the height h of the water was given by h ( t )= 5 + 4.9 cos . Approximate the time periods in the next 24
6
hours when the height of the water was above 6 feet.
Chapter 1 Review - P. 76 #1,2,4,5ab,6,7,10,14abcde,17a,18,19a,20,22,26,29 – Calculator Required
1. True/False
a. log 45 − 2 log 3 =
log 5
b. The function f defined by f ( x ) = −2 sin
x
has amplitude 2 and period 6π .
3
c. f ( x ) = x 2 − 2x, then f ( a + 1) = a2 + 3 .
d. If the function g is odd and g(-2) = 5, then g(2) = -5.
ln
1
2
7
− e3ln 2 =
2
2x − 3 and g ( x ) =
log2 x, evaluate
2. Given f ( x ) =
e. e
a. f g ( 4 ) 
b. g  f ( 2 ) 
c. f g−1 ( 0 ) 
d. g  f −1 (1) 




4. The order in which the transformations are applied to the graph of the given function is specified. Given an equation
for the transformed function in each case.
a. f ( x ) = x ; vertical stretch 3 and shift up 4.
1
, shift left 2 and shift down 1.
2
c. h ( x ) = e x ; reflect through x-axis, shift right 1 and shift down 1.
b. g ( x ) = ln x ; vertical shrink by
5. The function f is defined by f ( x ) =
x −1
.
2x 2 − 8
a. Determine the x- and y-intercepts, if any.
b. Write equations for any horizontal and vertical asymptotes.
6. Given f ( x ) =x3 − 6x 2 + 9x and g ( x ) =4
a. Find coordinates of the points common to the graphs of f and g.
b. Find all the zeros of f.
c. If the domain of f is limited to the closed interval [0, 2], what is the range of f?
7. Let f be the function given by f ( x ) =
a.
b.
c.
d.
3x
x2 − x + 1
.
Find the domain of f.
Sketch a graph of f on -5 < x < 5 and −4 ≤ y ≤ 4 .
Write an equation for each horizontal asymptote of the graph of f.
Find the range of f.
10.
4
a. Plot the graphs
of y x=
=
and y 3 x . Determine the points where they intersect correct to three decimal places.
b. For what values of x is 3 x > x 4 ?
14. Each function below has the interval [-5, 5] as its domain.
a.
b.
c.
d.
Specify the range of f and g.
For what inputs x does f(x) = g(5)?
List all intervals (approximately) on which the function f is increasing.
Evaluate, if possible:
f (3)
ii) 2f ( 2 ) − g ( −2 )
iii)
i) f ( −3 ) ⋅ g ( 0 )
g ( −2 )
e. Evaluate, if possible:
i) f g ( 4 ) 
17.
ii) g  f ( 4 ) 
iii) f  f ( −4 ) 
a. Let the function f be defined by the formula f ( x=
) x2 −
x
g(x)
Evaluate, if possible:
i) f(1) + g(1)
-3
-4
-2
-9
-1
-1
ii) f ( 3 ) ⋅ g ( 3 )
18. If g =
( x ) 2x 2 + 3x, determine and simplify: i) g(7.13)
0
2
iii)
iv)
f ( −3 )
g ( −4 )
v)
f (3)
g ( −3 )
iv) g g ( 2 ) 
3
and let the function g be defined by the following table:
x
1
2
3
4
0
8
3
-2
f (0)
g (0)
iv) f g ( 0 ) 
ii) g(2 + h) – g(2)
3x
, find a rule for f −1 ( x ) .
5+x
20. Sketch the graph of a function that is continuous on the domain 0 < x < 10 and has all of the following properties:
i) the range of f is the interval [0, 5]
ii) f(0) = 1
iii) the graph of f is concave up on the interval (0, 4)
iv) the graph of f is concave down on the interval (4, 10).
19. a. If f ( x ) =
22. Give an example of a rational function that has vertical asymptotes at x = -1 and x = 2 and that has the line y = 3 as a
horizontal asymptote.
26. Sketch the graph of an even rational function with all of the following properties:
i) zeros at x = 1 and x = -1;
ii) vertical asymptotes at x = 3 and x = -3
iii) a horizontal asymptote of y = 1;
1
iv) f ( 0 ) =
9
29. Values of the functions f and g for several arguments x are given in the table below.
x
1
2
3
4
5
6
7
f(x)
10
9
8
7
6
5
4
g(x)
5
6
7
8
7
6
5
1
. Use the information about f and g given in the table and the definition of h to evaluate the
Let h ( x ) =
2
x +1
following.
f 
b.   ( 4 )
c. [ f  g] ( 2 )
d. h  f ( 7 ) 
e. [ f  g  h] ( 0 )
a. [ f + g] (1)
g