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MAT 153: College Algebra and Trigonometry

Subsection 2.6 Functions and Inverse Functions Exercises

  1. Refer to the function \(g(x)=7x+2\text{.}\)
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    1. What are the domain and range of \(g\text{?}\)
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    2. Let \(h(x)=x^2+x+1\text{.}\) Find an algebraic expression for \(g(h(x))\text{.}\)
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    3. Again using \(h(x)=x^2+x+1\text{,}\) find an algebraic expression for \(3g(x) - h(x)\text{.}\)
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    4. Find a function equation for the inverse function, \(g^{-1}\text{.}\)
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    5. What are the domain and range of \(g^{-1}\text{?}\)
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  2. Refer to the graph of \(R(x)\) in Figure 2.6.13. The diagonal line \(y=x\) is included on the graph for reference. Note that the graph of \(R(x)\) includes the points \((-7.5,0)\text{,}\) \((-1,2)\text{,}\) \((0,4.5)\text{,}\) and \((6,6)\text{.}\)
    A graph showing a nonlinear increasing function in red and the dashed diagonal line y equals x in blue.
    Figure 2.6.13.
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    1. What are the domain and range of \(R\text{?}\)
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    2. Draw the graph of the inverse function, \(R^{-1}\text{.}\)
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    3. What are the domain and range of \(R^{-1}\text{?}\)
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  3. Refer to \(f\) in Table 2.6.14.
    Table 2.6.14. \(f(x)\)
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    \(x\) 0 1 2 3
    \(f(x)\) 3 0 2 -4
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    1. What are the domain and range of \(f\text{?}\)
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    2. Find a function table for the inverse function, \(f^{-1}\text{.}\)
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    3. What are the domain and range of \(f^{-1}\text{?}\)
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  4. Let \(A(x)=4(x-6)^{\frac{3}{2}}\text{.}\)
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    1. Write functions \(b(x)\) and \(c(x)\) so that \(A(x)=b(c(x))\text{.}\)
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    2. What are the domain and range of \(A(x)\text{?}\)
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    3. Find a function equation for the inverse function, \(A^{-1}(x)\text{.}\)
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    4. What are the domain and range of \(A^{-1}(x)\text{?}\)
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    5. On what interval(s) is \(A\) increasing? On what interval(s) is \(A\) decreasing?
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  5. Let \(Z(x)=(x^2+4)^{\frac{1}{2}}\text{,}\) \(x \geq 0\text{.}\)
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    1. True or false: For all \(x \geq 0\text{,}\) \(Z(x)=x+2\text{.}\) Explain your answer.
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    2. What are the domain and range of \(Z(x)\text{?}\)
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    3. Find a function equation for the inverse function, \(Z^{-1}(x)\text{.}\)
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    4. What are the domain and range of \(Z^{-1}(x)\text{?}\)
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  6. Refer to \(Q(x)=\sqrt[3]{x-3}\text{.}\)
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    1. What are the domain and range of \(Q\text{?}\)
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    2. Let \(U(x)=x^3\text{.}\) Find an algebraic expression for \(U(Q(x))\text{.}\)
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    3. Find a function equation for the inverse function, \(Q^{-1}\text{.}\)
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    4. What are the domain and range of \(Q^{-1}\text{?}\)
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  7. The Jaguar XJ AWD loses value more rapidly than some of its competitors. From the initial purchase price of $76,700, an owner can expect the value of the car 5 years later to be $52,014.
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    1. What is the average dollar value decline per year during the first 5 years of ownership?
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    2. Use an exponential model produce a function that gives the value of the Jaguar in terms of the number of years, \(t\text{,}\) since the car was new.
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    3. Use your model to predict the value of the Jaguar at 3 years old.
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    4. When will the value of the Jaguar decline to $30,000?
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  8. Simplify each expression below and write without negative exponents.
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    1. \(\displaystyle \frac{xy^7}{x^3y^4}\)
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    2. \(\displaystyle \left(\frac{2X^3}{-8X^4}\right)^2\)
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    3. \(\displaystyle \frac{8x^3(x^2)^8}{(-2x^6y)^2}\)
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