College Algebra
Chapter 4.1 – 4.5 Test A
Name
Period:______
Directions: Show all work and reasoning to receive full credit.
1) Determine the composite function ( f g )( x) given the functions 𝑓(𝑥) = 3𝑥 2 + 2𝑥 + 1
and 𝑔(𝑥) = 2𝑥 − 1.
2) Determine
 f  g  2 given the functions
f ( x)   3 x  15 and g ( x)  2 x 2  x .
3) Determine algebraically if the given functions are inverses.
1
2
𝑓(𝑥) =
𝑎𝑛𝑑 ℎ(𝑥) = −
𝑥+3
𝑥
4) The functions f (x) and g (x ) are inverses of each other. Completely answer and explain the following:
a) State what is true about the graphs of f (x) and g (x ) .
b) State what is true about the compositions of f (x) and g (x ) . In other words, what is true about ( f  g )( x)
and ( g  f )( x) ?
5) Use the given function to answer the following questions.
3𝑥 + 2
ℎ(𝑥) =
2𝑥 − 1
a) Determine the inverse of the function.
b) Determine the domain and range of ℎ(𝑥).
c) Determine the domain and range of ℎ−1 (𝑥).
d) Verify algebraically that your inverse is correct.
4
6) The volume of a hot air balloon as a function of radius, r, is given by 𝑉(𝑟) = 3 𝜋𝑟 3 . Find the volume of the
3
balloon as a function of time if the radius with time according to the function 𝑟(𝑡) = 4 𝑡 3 .
7) If 2−2𝑥 = 4 what does 82𝑥 equal?
8) Solve the following equations algebraically.
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a) (𝑒 4 )𝑥 ∙ 𝑒 𝑥 = 𝑒 12
−5𝑥
c) 4
∙4
𝑥+1
1
= 64
b) 26𝑥−3 ∙ 22𝑥+1 = 24𝑥
d)
𝑥
𝑥
27 ∙ 9 =
32𝑥+1
9
e) log
1
1000
= 7𝑥 − 3
g) log 3 (𝑥 2 + 10𝑥 − 105) = 4
f) log 5 625 = 2𝑥 − 16
9) Write as a sum and/or difference of logarithms. Express all powers as factors, factor completely, and
combine like terms.
ln
(𝑥 2 − 4)
√𝑥
1
10) Condense the logarithm by writing it as a single logarithm: 2 log 𝑥 + log(2𝑥 + 1) − 3 log 𝑥 − log(𝑥 + 1).