Stewart Precalc 6e Section 2.6: Combining Functions

1 Concepts - Combining functions from graph · Level 1

From the graphs of $f$ and $g$ in the figure, find (a) $(f+g)(2)$, (b) $(f-g)(2)$, (c) $(f g)(2)$, (d) $\left(\dfrac{f}{g}\right)(2)$.

2 Concepts - Composition definition · Level 1

By definition, $(f \circ g)(x) = $ ____. So if $g(2) = 5$ and $f(5) = 12$, then $(f \circ g)(2) = $ ____.

3 Concepts - Composition rules · Level 1

If the rule of the function $f$ is 'add one' and the rule of the function $g$ is 'multiply by 2,' then state the rule of $f \circ g$ and the rule of $g \circ f$.

4 Concepts - Composition algebra · Level 1

Express the functions in Exercise 3 algebraically: $f(x) = $ ____, $g(x) = $ ____, $(f \circ g)(x) = $ ____, $(g \circ f)(x) = $ ____.

5 Skills - Sum, difference, product, quotient · Level 2

Find $f + g$, $f - g$, $f g$, and $\dfrac{f}{g}$ and their domains for $f(x) = x - 3$, $g(x) = x^2$.

6 Skills - Sum, difference, product, quotient · Level 2

Find $f + g$, $f - g$, $f g$, and $\dfrac{f}{g}$ and their domains for $f(x) = x^2 + 2x$, $g(x) = 3x^2 - 1$.

7 Skills - Sum, difference, product, quotient · Level 2

Find $f + g$, $f - g$, $f g$, and $\dfrac{f}{g}$ and their domains for $f(x) = \sqrt{4 - x^2}$, $g(x) = \sqrt{1 + x}$.

8 Skills - Sum, difference, product, quotient · Level 2

Find $f + g$, $f - g$, $f g$, and $\dfrac{f}{g}$ and their domains for $f(x) = \sqrt{9 - x^2}$, $g(x) = \sqrt{x^2 - 4}$.

9 Skills - Sum, difference, product, quotient · Level 2

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

10 Skills - Sum, difference, product, quotient · Level 2

Find $f + g$, $f - g$, $f g$, and $\dfrac{f}{g}$ and their domains for $f(x) = 2/(x + 1)$, $g(x) = x/(x + 1)$.

11 Skills - Domain · Level 2

Find the domain of the function $f(x) = \sqrt{x} + \sqrt{1 - x}$.

12 Skills - Domain · Level 2

Find the domain of the function $g(x) = \sqrt{x + 1} - \dfrac{1}{x}$.

13 Skills - Domain · Level 2

Find the domain of the function $h(x) = (x - 3)^{-\dfrac{1}{4}}$.

14 Skills - Domain · Level 2

Find the domain of the function $k(x) = \dfrac{\sqrt{x + 3}}{x - 1}$.

15 Skills - Graphical addition · Level 2

Use graphical addition to sketch the graph of $f + g$ from the figure.

16 Skills - Graphical addition · Level 2

Use graphical addition to sketch the graph of $f + g$ from the figure.

17 Skills - Graph addition with formula · Level 2

Draw the graphs of $f$, $g$, and $f + g$ on a common screen to illustrate graphical addition for $f(x) = \sqrt{1 + x}$, $g(x) = \sqrt{1 - x}$.

18 Skills - Graph addition with formula · Level 2

Draw the graphs of $f$, $g$, and $f + g$ on a common screen for $f(x) = x^2$, $g(x) = \sqrt{x}$.

19 Skills - Graph addition with formula · Level 2

Draw the graphs of $f$, $g$, and $f + g$ on a common screen for $f(x) = x^2$, $g(x) = \left(\dfrac{1}{3}\right) x^3$.

20 Skills - Graph addition with formula · Level 2

Draw the graphs of $f$, $g$, and $f + g$ on a common screen for $f(x) = \sqrt[4]{1 - x}$, $g(x) = \sqrt{1 - x^2/9}$.

21 Skills - Composition evaluation · Level 2

Use $f(x) = 3x - 5$ and $g(x) = 2 - x^2$ to evaluate (a) $f(g(0))$, (b) $g(f(0))$.

22 Skills - Composition evaluation · Level 2

Use $f(x) = 3x - 5$ and $g(x) = 2 - x^2$ to evaluate (a) $f(f(4))$, (b) $g(g(3))$.

23 Skills - Composition evaluation · Level 2

Use $f(x) = 3x - 5$ and $g(x) = 2 - x^2$ to evaluate (a) $(f \circ g)(-2)$, (b) $(g \circ f)(-2)$.

24 Skills - Composition evaluation · Level 2

Use $f(x) = 3x - 5$ and $g(x) = 2 - x^2$ to evaluate (a) $(f \circ f)(-1)$, (b) $(g \circ g)(2)$.

25 Skills - Composition formula · Level 3

Use $f(x) = 3x - 5$ and $g(x) = 2 - x^2$ to find (a) $(f \circ g)(x)$, (b) $(g \circ f)(x)$.

26 Skills - Composition formula · Level 3

Use $f(x) = 3x - 5$ and $g(x) = 2 - x^2$ to find (a) $(f \circ f)(x)$, (b) $(g \circ g)(x)$.

27 Skills - Composition from graphs · Level 2

Use the given graphs of $f$ and $g$ to evaluate $f(g(2))$.

28 Skills - Composition from graphs · Level 2

Use the given graphs of $f$ and $g$ to evaluate $g(f(0))$.

29 Skills - Composition from graphs · Level 2

Use the given graphs of $f$ and $g$ to evaluate $(g \circ f)(4)$.

30 Skills - Composition from graphs · Level 2

Use the given graphs of $f$ and $g$ to evaluate $(f \circ g)(0)$.

31 Skills - Composition from graphs · Level 2

Use the given graphs of $f$ and $g$ to evaluate $(g \circ g)(-2)$.

32 Skills - Composition from graphs · Level 2

Use the given graphs of $f$ and $g$ to evaluate $(f \circ f)(4)$.

33 Skills - Find compositions · Level 2

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = 2x + 3$, $g(x) = 4x - 1$.

34 Skills - Find compositions · Level 2

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = 6x - 5$, $g(x) = \dfrac{x}{2}$.

35 Skills - Find compositions · Level 2

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

36 Skills - Find compositions · Level 2

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = x^3 + 2$, $g(x) = \sqrt[3]{x}$.

37 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = \dfrac{1}{x}$, $g(x) = 2x + 4$.

38 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = x^2$, $g(x) = \sqrt{x - 3}$.

39 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = |x|$, $g(x) = 2x + 3$.

40 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = x - 4$, $g(x) = |x + 4|$.

41 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = x/(x + 1)$, $g(x) = 2x - 1$.

42 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = \dfrac{1}{\sqrt{x}}$, $g(x) = x^2 - 4x$.

43 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = x/(x + 1)$, $g(x) = \dfrac{1}{x}$.

44 Skills - Find compositions · Level 3

Find $f \circ g$, $g \circ f$, $f \circ f$, and $g \circ g$, and their domains, for $f(x) = \dfrac{2}{x}$, $g(x) = x/(x + 2)$.

45 Skills - Triple composition · Level 3

Find $f \circ g \circ h$ for $f(x) = x - 1$, $g(x) = \sqrt{x}$, $h(x) = x - 1$.

46 Skills - Triple composition · Level 3

Find $f \circ g \circ h$ for $f(x) = \dfrac{1}{x}$, $g(x) = x^3$, $h(x) = x^2 + 2$.

47 Skills - Triple composition · Level 3

Find $f \circ g \circ h$ for $f(x) = x^4 + 1$, $g(x) = x - 5$, $h(x) = \sqrt{x}$.

48 Skills - Triple composition · Level 3

Find $f \circ g \circ h$ for $f(x) = \sqrt{x}$, $g(x) = x/(x - 1)$, $h(x) = \sqrt[3]{x}$.

49 Skills - Express as composition · Level 2

Express the function $F(x) = (x - 9)^5$ in the form $f \circ g$.

50 Skills - Express as composition · Level 2

Express the function $F(x) = \sqrt{x} + 1$ in the form $f \circ g$.

51 Skills - Express as composition · Level 2

Express the function $G(x) = x^2/(x^2 + 4)$ in the form $f \circ g$.

52 Skills - Express as composition · Level 2

Express the function $G(x) = 1/(x + 3)$ in the form $f \circ g$.

53 Skills - Express as composition · Level 2

Express the function $H(x) = |1 - x^3|$ in the form $f \circ g$.

54 Skills - Express as composition · Level 2

Express the function $H(x) = \sqrt{1 + \sqrt{x}}$ in the form $f \circ g$.

55 Skills - Express as triple composition · Level 3

Express the function $F(x) = 1/(x^2 + 1)$ in the form $f \circ g \circ h$.

56 Skills - Express as triple composition · Level 3

Express the function $F(x) = \sqrt[3]{\sqrt{x} - 1}$ in the form $f \circ g \circ h$.

57 Skills - Express as triple composition · Level 3

Express the function $G(x) = (4 + \sqrt[3]{x})^9$ in the form $f \circ g \circ h$.

58 Skills - Express as triple composition · Level 3

Express the function $G(x) = 2/(3 + \sqrt{x})^2$ in the form $f \circ g \circ h$.

59 Applications - Revenue · Level 3

A print shop makes bumper stickers for election campaigns. If $x$ stickers are ordered (where $x < 10000$), then the price per bumper sticker is $0.15 - 0.000002 x$ dollars, and the total cost of producing the order is $0.095 x - 0.0000005 x^2$ dollars. Use the fact that revenue $=$ price per item $\times$ number of items sold to express $R(x)$, the revenue from an order of $x$ stickers, as a product of two functions of $x$.

60 Applications - Profit · Level 3

Use the fact that profit $=$ revenue $-$ cost to express $P(x)$, the profit on an order of $x$ stickers, as a difference of two functions of $x$. (See Exercise 59.)

61 Applications - Area of a Ripple · Level 3

A stone is dropped in a lake, creating a circular ripple that travels outward at a speed of $60$ cm/s. (a) Find a function $g$ that models the radius as a function of time. (b) Find a function $f$ that models the area of the circle as a function of the radius. (c) Find $f \circ g$. What does this function represent?

62 Applications - Inflating a Balloon · Level 3

A spherical balloon is being inflated. The radius of the balloon is increasing at the rate of $1$ cm/s. (a) Find a function $f$ that models the radius as a function of time. (b) Find a function $g$ that models the volume as a function of the radius. (c) Find $g \circ f$. What does this function represent?

63 Applications - Area of a Balloon · Level 3

A spherical weather balloon is being inflated. The radius of the balloon is increasing at the rate of $2$ cm/s. Express the surface area of the balloon as a function of time $t$ (in seconds).

64 Applications - Multiple Discounts · Level 3

You have a $\$50$ coupon from the manufacturer good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a $20%$ discount on all cell phones. Let $x$ represent the regular price of the cell phone. (a) Suppose only the $20%$ discount applies. Find a function $f$ that models the purchase price. (b) Suppose only the $\$50$ coupon applies. Find a function $g$ that models the purchase price. (c) Find both $(f \circ g)(x)$ and $(g \circ f)(x)$. Which composition gives the lower price?

65 Applications - Multiple Discounts · Level 3

An appliance dealer advertises a $10%$ discount on all his washing machines. In addition, the manufacturer offers a $\$100$ rebate on the purchase of a washing machine. Let $x$ represent the sticker price. (a) Find $f$ for the $10%$ discount only. (b) Find $g$ for the $\$100$ rebate only. (c) Find $f \circ g$ and $g \circ f$. Which is the better deal?

66 Applications - Airplane Trajectory · Level 3

An airplane is flying at a speed of $350$ mi/h at an altitude of one mile. The plane passes directly above a radar station at time $t = 0$. (a) Express the distance $s$ (in miles) between the plane and the radar station as a function of the horizontal distance $d$. (b) Express $d$ as a function of time $t$ (in hours). (c) Use composition to express $s$ as a function of $t$.

67 Discovery - Compound Interest · Level 4

A savings account earns $5%$ interest compounded annually. If you invest $x$ dollars, then $A(x) = x + 0.05 x = 1.05 x$ is the amount after one year. Find $A \circ A$, $A \circ A \circ A$, and $A \circ A \circ A \circ A$. What do these compositions represent? Find a formula for the composition of $n$ copies of $A$.

68 Discovery - Composing Linear Functions · Level 4

The graphs of $f(x) = m_1 x + b_1$ and $g(x) = m_2 x + b_2$ are lines with slopes $m_1$ and $m_2$, respectively. Is the graph of $f \circ g$ a line? If so, what is its slope?

69 Discovery - Solving for Unknown Function · Level 4

Suppose $g(x) = 2x + 1$ and $h(x) = 4x^2 + 4x + 7$. Find a function $f$ such that $f \circ g = h$. Now suppose $f(x) = 3x + 5$ and $h(x) = 3x^2 + 3x + 2$. Find a function $g$ such that $f \circ g = h$.

70 Discovery - Odd and Even Compositions · Level 4

Suppose $h = f \circ g$. (a) If $g$ is even, is $h$ necessarily even? (b) If $g$ is odd, is $h$ odd? (c) What if $g$ is odd and $f$ is odd? (d) What if $g$ is odd and $f$ is even?

Quick Answer Input

Submit your exam?