College Algebra Problems With Answers sample 2 : Composite and Inverse Functions
College algebra problems and questions on composite and inverse functions are presented along with their solutions located at the bottom of the page.
Problems
Let \( f(x) = \sqrt{x - 4} + 3 \).
a) Find the inverse of \( f \).
b) Find the range of \( f^{-1} \).
Let \( h(x) = \frac{x - 1}{-x + 3} \).
a) Find the inverse of \( h \).
b) Find the range of \( h \).
Let \( f(x) = \frac{x - 1}{x + 5} \) and \( g(x) = \frac{1}{x + 3} \).
a) Find the composite function \( (f \circ g)(x) \).
b) Find the domain of \( f \circ g \).
Function \( f \) is a function with inverse \( f^{-1} \). Function \( h \) is defined by \( h(x) = f(x) + k \) where \( k \) is a constant. Express the inverse function of \( h \) in terms of \( f^{-1} \) and \( k \).
Function \( f \) is a function with inverse \( f^{-1} \). Function \( h \) is defined by \( h(x) = Af(x - h) + k \) where \( A \), \( k \), and \( h \) are constants. Express the inverse function of \( h \) in terms of \( f^{-1} \), \( A \), \( k \), and \( h \).
The graphs of functions \( f \) and \( g \) are shown below.
a) Use the graph to find \( (f \circ g)(-4) \).
b) Use the graph to find \( (g \circ f)(1) \).
Functions \( f \) and \( h \) are defined by the tables
x
-3
-2
-1
0
1
2
3
\( f(x) \)
-6
-4
-2
1
2
6
16
x
0
1
2
3
4
5
6
\( h(x) \)
1
2
5
10
17
26
37
Use the values in the tables to find
a) \( (f \circ h)(1) \)
b) \( (f \circ f)(0) \)
c) \( (f \circ h)(5) \)
d) \( (f \circ h^{-1})(5) \)
e) \( (h \circ f^{-1})(6) \)
Solutions to the Above Problems
\( f^{-1}(x) = (x - 3)^3 + 4 \), \( x \geq 3 \)
\([4, +\infty)\): it is the domain of \( f \)
\( h^{-1}(x) = \frac{-3x - 1}{x + 1} \)
\((-\infty, -1) \cup (-1, +\infty)\): it is the domain of \( h^{-1} \)
\( (f \circ g)(x) = -\frac{x + 2}{5x + 16} \)
Domain of the composite function \( f \circ g \): \((- \infty, -\frac{16}{5}) \cup (-\frac{16
}{5}, -3) \cup (-3, +\infty)\)