# 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

1. Let $$f(x) = \sqrt{x - 4} + 3$$.
a) Find the inverse of $$f$$.
b) Find the range of $$f^{-1}$$.

2. Let $$h(x) = \frac{x - 1}{-x + 3}$$.
a) Find the inverse of $$h$$.
b) Find the range of $$h$$.

3. 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 $$f \circ g$$.

4. 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 $$k$$.

5. 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 $$h$$.

6. 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 $$(g \circ f)(1)$$.

7. 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

1. $$f^{-1}(x) = (x - 3)^3 + 4$$, $$x \geq 3$$

2. $$[4, +\infty)$$: it is the domain of $$f$$

1. $$h^{-1}(x) = \frac{-3x - 1}{x + 1}$$

2. $$(-\infty, -1) \cup (-1, +\infty)$$: it is the domain of $$h^{-1}$$

1. $$(f \circ g)(x) = -\frac{x + 2}{5x + 16}$$

2. Domain of the composite function $$f \circ g$$: $$(- \infty, -\frac{16}{5}) \cup (-\frac{16 }{5}, -3) \cup (-3, +\infty)$$

1. $$h^{-1}(x) = f^{-1}(x - k)$$

2. $$h^{-1}(x) = f^{-1}(\frac{x - k}{A}) + h$$

1. $$(f \circ g)(-4) = f(g(-4)) = f(2) = -2$$

2. $$(g \circ f)(1) = g(f(1)) = g(-3) = -1$$

1. $$(f \circ h)(1) = f(h(1)) = f(2) = 6$$

2. $$(f \circ f)(0) = f(f(0)) = f(1) = 2$$

3. $$(f \circ h)(5) = f(h(5)) = f(26) = \text{undefined}$$

4. $$(f \circ h^{-1})(5) = f(h^{-1}(5)) = f(2) = 6$$

5. $$(h \circ f^{-1})(6) = h(f^{-1}(6)) = h(2) = 5$$