Stewart Precalc 6e Section 2.Review

1 Average Rate of Change - Linear · Level 2

For \(f(x) = \dfrac{1}{2} x - 6\): (a) Find the average rate of change of \(f\) between \(x = 0\) and \(x = 2\), and between \(x = 15\) and \(x = 50\). (b) Were the two average rates of change you found in (a) the same? Explain why or why not.

2 Average Rate of Change - Linear · Level 2

For \(f(x) = 8 - 3 x\): (a) Find the average rate of change of \(f\) between \(x = 0\) and \(x = 2\), and between \(x = 15\) and \(x = 50\). (b) Were the two average rates of change the same? Explain.

3 Transformations · Level 2

Suppose the graph of \(f\) is given. Describe how the graphs of the following functions can be obtained from the graph of \(f\). (a) \(y = f(x) + 8\) (b) \(y = f(x + 8)\) (c) \(y = 1 + 2 f(x)\) (d) \(y = f(x - 2) - 2\) (e) \(y = f(-x)\) (f) \(y = -f(-x)\) (g) \(y = -f(x)\) (h) \(y = f^{-1}(x)\)

4 Transformations · Level 2

The graph of \(f\) is given. Draw the graphs of the following functions. (a) \(y = f(x - 2)\) (b) \(y = -f(x)\) (c) \(y = 3 - f(x)\) (d) \(y = \dfrac{1}{2} f(x) - 1\) (e) \(y = f^{-1}(x)\) (f) \(y = f(-x)\)

5 Even/Odd Functions · Level 2

Determine whether \(f\) is even, odd, or neither. (a) \(f(x) = 2 x^5 - 3 x^2 + 2\) (b) \(f(x) = x^3 - x^7\) (c) \(f(x) = \dfrac{1 - x^2}{1 + x^2}\) (d) \(f(x) = \dfrac{1}{x + 2}\)

6 Even/Odd from Graph · Level 2

Determine whether the function in each figure is even, odd, or neither.

7 Extrema · Level 2

Find the minimum value of \(g(x) = 2 x^2 + 4 x - 5\).

8 Extrema · Level 2

Find the maximum value of \(f(x) = 1 - x - x^2\).

9 Applications - Extrema · Level 3

A stone is thrown upward from the top of a building. Its height (in feet) above the ground after \(t\) seconds is \(h(t) = -16 t^2 + 48 t + 32\). What maximum height does it reach?

10 Local Extrema - Calculator · Level 3

Find the local maximum and minimum values of \(f(x) = 3.3 + 1.6 x - 2.5 x^3\) and the values of \(x\) at which they occur. State each answer correct to two decimal places.

11 Local Extrema - Calculator · Level 3

Find the local maximum and minimum values of \(f(x) = x^{\dfrac{2}{3}} (6 - x)^{\dfrac{1}{3}}\) and the values of \(x\) at which they occur. State each answer correct to two decimal places.

12 Graphical Addition · Level 2

For \(f(x) = x + 2\) and \(g(x) = x^2\), draw the graphs of \(f\), \(g\), and \(f + g\) on the same axes to illustrate graphical addition.

13 Graphical Addition · Level 2

For \(f(x) = x^2 + 1\) and \(g(x) = 3 - x^2\), draw the graphs of \(f\), \(g\), and \(f + g\) on the same axes.

14 Combinations of Functions · Level 3

If \(f(x) = x^2 - 3 x + 2\) and \(g(x) = 4 - 3 x\), find the following functions. (a) \(f + g\) (b) \(f - g\) (c) \(f g\) (d) \(\dfrac{f}{g}\) (e) \(f \circ g\)

15 Composition · Level 3

If \(f(x) = 1 + x^2\) and \(g(x) = \sqrt{x - 1}\), find the following. (a) \(f \circ g\) (b) \(g \circ f\) (c) \((f \circ g)(2)\) (d) \((f \circ f)(2)\) (e) \(f \circ g \circ f\) (f) \(g \circ f \circ g\)

16 Composition · Level 3

Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\) and their domains for \(f(x) = 3 x - 1\), \(g(x) = 2 x - x^2\).

17 Composition · Level 3

Find \(f \circ g\), \(g \circ f\), \(f \circ f\), and \(g \circ g\) and their domains for \(f(x) = \sqrt{x}\), \(g(x) = \dfrac{2}{x - 4}\).

18 Composition · Level 3

Find \(f \circ g \circ h\), where \(f(x) = \sqrt{1 - x}\), \(g(x) = 1 - x^2\), and \(h(x) = 1 + \sqrt{x}\).

19 Decomposition · Level 3

If \(T(x) = \dfrac{1}{\sqrt{1 + \sqrt{x}}}\), find functions \(f\), \(g\), and \(h\) such that \(f \circ g \circ h = T\).

20 One-to-One Test · Level 2

Determine whether \(f(x) = 3 + x^3\) is one-to-one.

21 One-to-One Test · Level 2

Determine whether \(g(x) = 2 - 2 x + x^2\) is one-to-one.

22 One-to-One Test · Level 2

Determine whether \(h(x) = \dfrac{1}{x^4}\) is one-to-one.

23 One-to-One Test · Level 2

Determine whether \(r(x) = 2 + \sqrt{x + 3}\) is one-to-one.

24 One-to-One Test · Level 2

Determine whether \(p(x) = 3.3 + 1.6 x - 2.5 x^3\) is one-to-one.

25 One-to-One Test · Level 2

Determine whether \(q(x) = 3.3 + 1.6 x + 2.5 x^3\) is one-to-one.

26 Finding Inverses · Level 2

Find the inverse of \(f(x) = 3 x - 2\).

27 Finding Inverses · Level 2

Find the inverse of \(f(x) = \dfrac{2 x + 1}{3}\).

28 Finding Inverses · Level 2

Find the inverse of \(f(x) = (x + 1)^3\).

29 Finding Inverses · Level 2

Find the inverse of \(f(x) = 1 + \sqrt[5]{x - 2}\).

30 Restricted Domain Inverse · Level 3

(a) Sketch the graph of \(f(x) = x^2 - 4\), \(x \geq 0\). (b) Use part (a) to sketch the graph of \(f^{-1}\). (c) Find an equation for \(f^{-1}\).

31 Verifying One-to-One and Inverse · Level 3

(a) Show that the function \(f(x) = 1 + \sqrt[4]{x}\) is one-to-one. (b) Sketch the graph of \(f\). (c) Use part (b) to sketch the graph of \(f^{-1}\). (d) Find an equation for \(f^{-1}\).

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