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 \( (g \circ f)(1) \).
    college algebra problem 4, function f(x)

    Figure 1. Graph of Function \( f \)
    college algebra problem 4, function g(x) .

    Figure 2. Graph of Function \( g \)


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

More References and links

inverse function
composite function
Algebra Questions and problems
More ACT, SAT and Compass practice

{ezoic-ad-1}


Popular Pages



More Info

{ez_footer_ads}