Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 2
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41Worksheet # 1: Functions and inverse functions
1. Give the domain and ranges of the following functions.
+1
(a) f (x) =
+
2
1
(b) g(t) =
1
2
2. If f (x) = 5x + 7 and g(x) = x
, find f g and g f . Are the functions f g and g f the same function?
1
1
3. Let f (x) = 2 +
. Determine the inverse function of f , f
. Give the domain and range of f and
+3
1
1
the inverse function f
. Verify that f
f
(x) = x.
4. Consider the function whose graph appears below.
y=f(x)
1
1
(a) Find f (3), f
(2) and f
(f (2)).
1
(b) Give the domain and range of f and of f
.
1
(c) Sketch the graph of f
.
1
x
1
2
5. Let f (x) = x
+ 2x + 5. Find the largest value of a so that f is one to one on the interval (
, a]. Let
1
g be the function f with the domain (
, a]. Find the inverse function g
. Give the domain and
1
range of g
.
6. True or False:
(a) Every function has an inverse.
(b) If f
g(x) = x for all x in the domain of g, then f is the inverse of g.
(c) If f
g(x) = x for all x in the domain of g and g f (x) = x for all x in the domain of f , then f
is the inverse of g.
3
3
(d) If f (x) = 1/(x + 2)
and g is the inverse function of f , then g(x) = (x + 2)
.
(e) The function f (x) = sin(x) is one to one.
3
(f) The function f (x) = 1/(x + 2)
is one to one.
7. Find the slope, x-intercept, and y-intercept of the line 3x
2y = 4.
1
8. Let f be a linear function with slope m with m = 0. What is the slope of the inverse function f
.
9. A ball is thrown in the air from ground level. The height of the ball in meters at time t seconds is
2
given by the function h(t) =
4.9t
+ 30t. At what time does the ball hit the ground (be sure to use
the proper units)?
10. We form a box by removing squares of side length x centimeters from the four corners of a rectangle of
width 100 cm and length 150 cm and then folding up the flaps between the squares that were removed.
a) Write a function which gives the volume of the box as a function of x. b) Give the domain for this
function.
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