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MAT 151: College Algebra with Trigonometry

Section 0.2 Review of Function Concepts

In MAT 103, you learned that functions can be represented in different ways, and that considering multiple function representations can give you different insights in solving problems.
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A triangular diagram illustrating the relationships among different representations of a function. The three nodes are labeled \(\text{Graph}\text{,}\) \(\text{Table}\text{,}\) and \(\text{Equation}\text{.}\) Arrows connect each node to the others, and the center of the triangle contains the text “Verbal Description,” indicating that verbal descriptions link all representations.
Figure 0.2.1. Diagram showing multiple representations of a function.
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You also learned that functions can be combined in different ways (using any of the above representations) to produce new functions.
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  • \(\displaystyle (f+g)(x)=f(x)+g(x)\)
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  • \(\displaystyle (f-g)(x)=f(x)-g(x)\)
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  • \(\displaystyle (f \cdot g)(x)=f(x) \cdot g(x)\)
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  • \(\displaystyle \left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}\)
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  • \(\displaystyle (f \circ g)(x)=f(g(x))\)
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Finally, you learned that a function \(f\) will have an inverse function \(f^{-1}\) only if the function is one-to-one (for each output there is only one input).
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We will continue to use these building blocks of functions as we explore different kinds of functions in MAT 151.
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Problem 0.2.2.

The distance above the ground of a ball tossed up vertically at a speed of 20 ft/s from the top of a building that is 50 feet tall is given by the equation
\begin{equation*} h(t)=-16t^2+20t+50 \end{equation*}
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  1. Sketch the graph of \(h\) on the axes below.
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    A blank coordinate grid labeled with \(h(t)\) on the vertical axis and \(t\) on the horizontal axis. The grid includes tick marks and numeric labels: the horizontal axis ranges from about -1 to 4, and the vertical axis ranges from 0 to 60 in increments of 5. No curve or data is plotted on the graph.
    Figure 0.2.3. Blank graph for plotting \(h(t)\text{.}\)
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  2. How long does it take the ball to hit the ground? Label the appropriate point on your graph from part 1.
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  3. What is the maximum height attained by the ball? At what time does it reach this height? Label the appropriate point on your graph from part 1.
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  4. At what time(s) is the ball at a height of 52 feet? Label the appropriate point(s) on your graph from part 1.
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  5. Solve the inequality \(h(t)>52\text{.}\)
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  6. What is the domain of \(h\) in the context of this problem? Write your answer in interval notation.
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  7. What is the range of \(h\) in the context of this problem? Write your answer in interval notation.
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  8. Is it appropriate to find an inverse function for \(h\text{?}\) Why or why not?
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Use the graph of \(f\) and the table for \(g\) shown below, and the function \(h(x)=\sqrt{x-2}+1\) to answer the following questions.
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The graph of \(f(x)\) consists of two separate curves. For \(x > 0\text{,}\) the curve starts at the origin and increases gradually, forming a concave-down shape. For \(x < 0\text{,}\) the curve decreases steeply and approaches negative infinity as \(x\) moves left, forming a concave-down shape. There is a filled circle at \((0, 0)\) and an open circle at \((0, -3)\text{,}\) indicating a jump discontinuity at \(x = 0\text{.}\)
Figure 0.2.4. Graph of \(f(x)\text{.}\)
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Table 0.2.5.
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\(x\) \(-4\) 0 4 8
\(g(x)\) \(-8\) \(-6\) \(-4\) \(-2\)
  1. Evaluate \(f(0)\text{.}\)
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  2. Solve \(f(x)=-6\text{.}\)
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  3. Solve \(f(x)=-1\text{.}\)
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  4. Solve \(g(x)=-4\text{.}\)
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  5. Solve \(h(x)=5\text{.}\)
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  6. Solve \(f(x) \geq 2\text{.}\)
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  7. Solve \(h(x) < 2\text{.}\)
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  8. Let \(j(x)=(g \circ f)(x)\text{.}\) Evaluate \(j(0)\text{.}\)
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  9. Let \(m(x)=2h(x)-g(x)\text{.}\) Evaluate \(m(4)\text{.}\)
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  10. Let \(n(x)=f(h(x))\text{.}\) Evaluate \(n(11)\text{.}\)
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  11. Solve \(n(x)=1\text{.}\)
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  12. What kind of function would you use to model the relationship shown in the table for \(g(x)\text{.}\) Why?
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  13. Write a function equation for \(g(x)\text{.}\)
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  14. Use your function equation from Item 13 to solve the \(g(x)=-5\text{.}\)
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  15. Write an equation for the inverse of \(g\text{,}\) \(g^{-1}\text{.}\)
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  16. What are the domain and range of \(h(x)\text{?}\)
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  17. Write a function equation for the inverse of \(h\text{,}\) \(h^{-1}\text{.}\)
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  18. What are the domain and range of \(h^{-1}\text{?}\)
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  19. What are the domain and range of \(f(x)\text{?}\)
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  20. What is the average rate of change of \(f\) on the interval \([0,4]\text{?}\)
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Problem 0.2.6.

Let \(y(x)=x^2-4x+3\text{,}\) \(z(x)=-2x+3\text{,}\) and \(w(x)=\frac{1}{x}\text{.}\) Use these function equations to answer each question below, showing your work algebraically.
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  1. Evaluate \(y(-3)\text{.}\)
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  2. Solve \(z(x)=-4\text{.}\)
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  3. Solve \(y(x)=3\text{.}\)
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  4. Solve \(w(x)=-2\text{.}\)
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  5. Solve \(z(x) \geq 5\text{.}\)
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  6. What is the vertical intercept of \(z(x)\text{?}\)
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  7. What is the \(x-\)intercept of \(z(x)\text{?}\)
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  8. What is the vertical intercept of \(y(x)\text{?}\)
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  9. What are the \(x-\)intercepts of \(y(x)\text{?}\)
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  10. Always/Sometimes/Never A function has a vertical intercept.
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  11. Always/Sometimes/Never A function has at least one \(x-\)intercept.
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  12. Solve \(y(x)=z(x)\text{.}\)
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  13. Solve \(y(x)>z(x)\text{.}\)
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  14. Solve \(z(x)=w(x)\text{.}\)
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  15. Let \(q(x)=y(x)-6z(x)\text{.}\) Write a function equation for \(q(x)\text{.}\)
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  16. Let \(v(x)=(y \circ z)(x)\text{.}\) Write a function equation for \(v(x)\text{.}\)
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  17. Solve \(v(x)=3\text{.}\)
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  18. Let \(r(x)=w(z(x))\text{.}\) Write a function equation for \(r(x)\text{.}\)
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  19. What are the domain and range of \(y(x)\text{?}\)
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  20. What is the domain of \(w(x)\text{?}\)
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