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MAT 151: College Algebra with Trigonometry

Subsection 1.1 Families of Functions Exercises

  1. For each function given in Item 1.e.i to Item 1.e.x:
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    1. Determine to which function family you think the function belongs.
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    2. Use Desmos to graph the function, and describe in words how it compares to the basic version of the function.
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    3. Describe the domain of the given function.
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    4. Describe the range of the given function.
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    5. State whether the function is even, odd, or neither even nor odd.
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      1. \(\displaystyle f(x)=x^2+4\)
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      2. \(\displaystyle g(x)=x^3-6x\)
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      3. \(\displaystyle h(x)=\sqrt{x-2}\)
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      4. \(\displaystyle k(x)=|x^2-4|\)
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      5. \(\displaystyle m(x)=\ln(x+1)+3\)
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      6. \(\displaystyle n(x)=\sqrt[3]{x^3-2x}\)
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      7. \(\displaystyle p(x)=x^2+\sqrt{x}\)
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      8. \(\displaystyle q(x)=x^3+x^{\frac{1}{3}}\)
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      9. \(\displaystyle r(x)=3^{2x}\)
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      10. \(\displaystyle s(x)=\frac{1}{x+7}\)
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  2. Refer to the function \(g(x)=\frac{1}{2}e^{.05x}\text{.}\)
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    1. Find a function equation for the inverse function, \(g^{-1}\text{.}\)
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    2. What are the domain and range of \(g^{-1}\text{?}\)
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    3. Graph \(g(x)\) and \(g^{-1}(x)\) on the same set of axes.
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    4. Solve \(g(x) \leq 2\text{.}\)
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  3. Refer to the population of Egypt, for given years as shown in Table 1.1.15.(Source: https://www.worldometers.info/world-population/egypt-population/)
    Table 1.1.15. Population, \(P(t)\) (in millions of people), of Egypt \(t\) years after 1985
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    \(t\) 0 5 10 15 20 25 30
    \(P(t)\) 50.2 57.4 63.7 69.9 76.8 84.1 93.8
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    1. Using the population of Egypt in 1985 and 2015, build an exponential model for the population, \(P(t)\text{,}\) \(t\) years after 1985.
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    2. Using the population of Egypt in 1985 and 2015, build a linear model for the population, \(P(t)\text{,}\) \(t\) years after 1985.
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    3. Use your exponential model to predict the population of Egypt in 2010. How does your prediction compare with the actual population at that time?
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    4. Use your linear model to predict the population of Egypt in 2010. How does your prediction compare with the actual population at that time?
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    5. Explain whether the linear or the exponential model is a better fit to the population data.
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    6. Use your exponential model to predict in what year the population of Egypt will reach 150 million. First give your answer in exact form using a logarithm, and then give the decimal approximation.
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    7. Use the exponential model \(P(t)\) to predict the population of Egypt in 2019. Compare your answer to the actual population of 101.2 million.
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  4. Solve the following equations.
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    1. \(\displaystyle 5\cdot 2^{3x}= 20480\)
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    2. \(\displaystyle 243^{6x}-151=578\)
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    3. \(\displaystyle 9^{2x}-23\cdot9^x - 108 = 0\)
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    4. \(\displaystyle \ln(x-1) + \ln(x-4) = \ln 70\)
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    5. \(\displaystyle 3^{3x}\cdot6^x=\frac{1}{2916}\)
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  5. Let \(f(x)=3^x\text{,}\) \(g(x)=15e^{3x}\text{,}\) and \(h(x)=\sqrt{x}\text{.}\)
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    1. Write an expression for \(f(x)g(x)\text{.}\)
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    2. Let \(k(x)=h(g(x))\text{.}\) Write an expression for \(k(x)\text{.}\)
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    3. Write an expression for \(g^{-1}(x)\text{,}\) the inverse of \(g(x)\text{.}\)
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    4. Find the exact solution to \(f(x)\geq g(x)\text{,}\) and express the solution using interval notation.
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