 # Intermediate Algebra: Functions and Graphs

## Section5.7Chapter 5 Summary and Review

### Subsection5.7.1Glossary

• function
• input variable
• output variable
• cube root
• absolute value
• proportional
• direct variation
• inverse variation
• constant of variation
• concavity
• scaling
• horizontal asymptote
• vertical asymptote

### Subsection5.7.2Key Concepts

1. A function can be described in words, by a table, by a graph, or by an equation.
2. #### Function Notation. 3. Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
4. The point $$(a, b)$$ lies on the graph of the function $$f$$ if and only if $$f(a)=b\text{.}$$
5. Each point on the graph of the function $$f$$ has coordinates $$(x, f(x))$$ for some value of $$x\text{.}$$
6. #### The Vertical Line Test.

A graph represents a function if and only if every vertical line intersects the graph in at most one point.
7. We can use a graphical technique to solve equations and inequalities.
8. $$b$$ is the cube root of $$a$$ if $$b$$ cubed equals $$a\text{.}$$ In symbols, we write
\begin{equation*} \blert{b=\sqrt{a}~~~~\text{if}~~~~b^3=a} \end{equation*}
9. #### Absolute Value.

The absolute value of $$x$$ is defined by
\begin{equation*} \abs{x} = \begin{cases} x \amp \text{if } x\ge 0\\ -x \amp \text{if } x\lt 0 \end{cases} \end{equation*}
10. Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.
11. The maximum or minimum of a quadratic function occurs at the vertex.
12. Eight basic functions and their graphs are important in applications:
\begin{gather*} f(x)=x~~~~~~f(x)=\abs{x}~~~~~~f(x)=x^2~~~~~~f(x)=x^3\\ f(x)=\sqrt{x}~~~~~f(x)=\sqrt{x}~~~~~~f(x)=\dfrac{1}{x}~~~~~~f(x)=\dfrac{1}{x^2} \end{gather*}
13. Two variables are directly proportional if the ratios of their corresponding values are always equal.
14. #### Direct Variation.

$$y$$ varies directly with $$x$$ if
\begin{equation*} y = kx \end{equation*}
where $$k$$ is a positive constant called the constant of variation.
15. Direct variation defines a linear function of the form
\begin{equation*} y = f (x) = kx \end{equation*}
The positive constant $$k$$ in the equation $$y = kx$$ is just the slope of the graph.
16. Direct variation has the following scaling property: increasing $$x$$ by any factor causes $$y$$ to increase by the same factor.
17. #### Direct Variation with a Power.

$$y$$ varies directly with a power of $$x$$ if
\begin{equation*} y = kx^n \end{equation*}
where $$k$$ and $$n$$ are positive constants.
18. If the ratio $$\dfrac{y}{x^n}$$ is constant, then $$y$$ varies directly with $$x^n\text{.}$$
19. #### Inverse Variation.

$$y$$ varies inversely with $$x$$ if
\begin{equation*} y = \dfrac{k}{x}\text{, }x \ne 0 \end{equation*}
where $$k$$ is a positive constant.
20. #### Inverse Variation with a Power.

$$y$$ varies inversely with $$x^n$$ if
\begin{equation*} y = \frac{k}{x^n}\text{, }x \ne 0 \end{equation*}
where $$k$$ and $$n$$ are positive constants.
21. If the product $$~yx^n~$$ is constant and $$n$$ is positive, then $$y$$ varies inversely with $$x^n\text{.}$$
22. A graph that bends upward is called concave up, and one that bends downward is concave down.

### Exercises5.7.3Chapter 5 Review Problems

#### Exercise Group.

Which of the tables in Problems 1–4 describe functions? Why or why not?
##### 1.
 $$x$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$3$$ $$y$$ $$6$$ $$0$$ $$1$$ $$2$$ $$6$$ $$8$$
A function: Each $$x$$ has exactly one associated $$y$$-value.
##### 2.
 $$p$$ $$3$$ $$-3$$ $$2$$ $$-2$$ $$-2$$ $$0$$ $$q$$ $$2$$ $$-1$$ $$4$$ $$-4$$ $$3$$ $$0$$
Not a function
##### 3.
 Student Score onIQ test Score onSAT test (A) $$118$$ $$649$$ (B) $$98$$ $$450$$ (C) $$110$$ $$590$$ (D) $$105$$ $$520$$ (E) $$98$$ $$490$$ (F) $$122$$ $$680$$
Not a function: The IQ of $$98$$ has two possible SAT scores.
##### 4.
 Student Correct answerson math quiz Quizgrade (A) $$13$$ $$85$$ (B) $$15$$ $$89$$ (C) $$10$$ $$79$$ (D) $$12$$ $$82$$ (E) $$16$$ $$91$$ (F) $$18$$ $$95$$
A function

#### 5.

The total number of barrels of oil pumped by the AQ oil company is given by the formula
\begin{equation*} N(t) = 2000 + 500t \end{equation*}
where $$N$$ is the number of barrels of oil $$t$$ days after a new well is opened. Evaluate $$N(10)$$ and explain what it means.
$$N(10) = 7000\text{:}$$ Ten days after the new well is opened, the company has pumped a total of $$7000$$ barrels of oil.

#### 6.

The number of hours required for a boat to travel upstream between two cities is given by the formula
\begin{equation*} H(v) = \dfrac{24}{v - 8} \end{equation*}
where $$v$$ represents the boat’s top speed in miles per hour. Evaluate $$H(16)$$ and explain what it means.
$$H(16) = 3\text{:}$$ At 16 mph, the trip takes 3 hours.

#### Exercise Group.

For Problems 7-10, evaluate the function for the given values.
##### 7.
$$F(t)=\sqrt{1+4t^2}\text{,}$$ $$~~F(0)~~$$ and $$~~F(-3)$$
$$F(0) = 1, ~~F(-3) =\sqrt{37}$$
##### 8.
$$G(x)=\sqrt{x-8}\text{,}$$ $$~~G(0)~~$$ and $$~~G(20)$$
$$G(0) = -2, ~~G(20) =\sqrt{12}$$
##### 9.
$$h(v)=6-\abs{4-2v} \text{,}$$ $$~~h(8)~~$$ and $$~~h(-8)$$
$$h(8) = -6, ~~h(-8) = -14$$
##### 10.
$$m(p)=\dfrac{120}{p+15} \text{,}$$ $$~~m(5)~~$$ and $$~~m(-40)$$
$$m(5) = 6, ~~m(-40) =-4.8$$

#### 11.

$$P(x)=x^2-6x+5$$
1. Compute $$P(0)\text{.}$$
2. Find all values of $$x$$ for which $$P(x)=0\text{.}$$
1. $$\displaystyle P(0)=5$$
2. x=5,~x=1

#### 12.

$$R(x)=\sqrt{4-x^2}$$
1. Compute $$R(0)\text{.}$$
2. Find all values of $$x$$ for which $$R(x)=0\text{.}$$
1. R(0)=2
2. $$\displaystyle x=2,~x=-2$$

#### Exercise Group.

For Problems 13 and 14, refer to the graphs to answer the questions.
##### 13.
1. Find $$f (-2)$$ and $$f (2)\text{.}$$
2. For what value(s) of $$t$$ is $$f (t) = 4\text{?}$$
3. Find the $$t$$- and $$f(t)$$-intercepts of the graph.
4. What is the maximum value of $$f\text{?}$$ For what value(s) of $$t$$ does $$f$$ take on its maximum value? 1. $$\displaystyle f (-2) = 3, ~~f (2) = 5$$
2. $$\displaystyle t = 1, ~~t = 3$$
3. $$t$$-intercepts $$(-3, 0), (4, 0)\text{;}$$ $$f (t)$$-intercept: $$(0, 2)$$
4. Maximum value of $$5$$ occurs at $$t = 2$$
##### 14.
1. Find $$P(-3)$$ and $$P(3)\text{.}$$
2. For what value(s) of $$z$$ is $$P(z) = 2\text{?}$$
3. Find the $$z$$- and $$P(z)$$-intercepts of the graph.
4. What is the minimum value of $$P\text{?}$$ For what value(s) of $$z$$ does $$P$$ take on its minimum value? 1. $$\displaystyle P(-3)=-2, ~~P(3)=3$$
2. $$\displaystyle z = -5,~ \dfrac{-1}{2},~4$$
3. $$(-4, 0), (-1, 0), (5,0)\text{;}$$ $$(0, 3)$$
4. Maximum value of $$-3$$ occurs at $$z = -2$$

#### Exercise Group.

Which of the graphs in Problems 15–18 represent functions?
##### 15. Function
##### 16. not a function
##### 17. Not a function
##### 18. Function

#### Exercise Group.

For Problems 19–22, graph the function by hand.
##### 19.
$$f(t)=-2t+4$$ ##### 20.
$$g(s)=\dfrac{-2}{3}s-2$$
##### 21.
$$p(x)=9-x^2$$ ##### 22.
$$q(x)=x^2-16$$

#### Exercise Group.

For Problems 23–26, graph the given function on a graphing calculator. Then use the graph to solve the equations and inequalities. Round your answers to one decimal place if necessary.
##### 23.
$$y=\sqrt{x}$$
1. Solve $$\sqrt{x} = 0.8$$
2. Solve $$\sqrt{x} = 1.5$$
3. Solve $$\sqrt{x}\gt 1.7$$
4. Solve $$\sqrt{x}\le 1.26$$
1. $$\displaystyle x = \dfrac{1}{2}= 0.5$$
2. $$\displaystyle x = \dfrac{27}{8}\approx 3.4$$
3. $$\displaystyle x \gt 4.9$$
4. $$\displaystyle x \le 2.0$$
##### 24.
$$y=\dfrac{1}{x}$$
1. Solve $$\dfrac{1}{x} = 2.5$$
2. Solve $$\dfrac{1}{x} = 0.3125$$
3. Solve $$\dfrac{1}{x}\ge 0.\overline{2}$$
4. Solve $$\dfrac{1}{x}\lt 5$$
1. $$\displaystyle x = 0.4$$
2. $$\displaystyle x = 3.2$$
3. $$\displaystyle 0 \lt x \le 4.5$$
4. $$x \lt 0$$ or $$x \gt 0.2$$
##### 25.
$$y=\dfrac{1}{x^2}$$
1. Solve $$\dfrac{1}{x^2} = 0.03$$
2. Solve $$\dfrac{1}{x^2} = 6.25$$
3. Solve $$\dfrac{1}{x^2}\gt 0.16$$
4. Solve $$\dfrac{1}{x^2}\le 4$$
1. $$\displaystyle x\approx\pm 5.8$$
2. $$\displaystyle x = \pm 0.4$$
3. $$-2.5\lt x \lt 0$$ or $$0\lt x\lt 2.5$$
4. $$x \le -0.5$$ or $$x\ge 0.5$$
##### 26.
$$y=\sqrt{x}$$
1. Solve $$\sqrt{x} = 0.707$$
2. Solve $$\sqrt{x} = 1.7$$
3. Solve $$\sqrt{x}\lt 1.5$$
4. Solve $$\sqrt{x}\ge 1.3$$
1. $$\displaystyle x = 0.5$$
2. $$\displaystyle x = 2.9$$
3. $$\displaystyle 0 \le x \lt 2.3$$
4. $$\displaystyle x \ge 1.7$$

#### Exercise Group.

In Problems 27–30, $$y$$ varies directly or inversely with a power of $$x\text{.}$$ Find the power of $$x$$ and the constant of variation, $$k\text{.}$$ Write a formula for each function of the form $$y = kx^n$$ or $$y = \dfrac{k}{x^n}\text{.}$$
##### 27.
 $$x$$ $$y$$ $$2$$ $$4.8$$ $$5$$ $$30.0$$ $$8$$ $$76.8$$ $$11$$ $$145.2$$
$$y = 1.2x^2$$
##### 28.
 $$x$$ $$y$$ $$1.4$$ $$75.6$$ $$2.3$$ $$124.2$$ $$5.9$$ $$318.6$$ $$8.3$$ $$448.2$$
$$y = 54x$$
##### 29.
 $$x$$ $$y$$ $$0.5$$ $$40.0$$ $$2.0$$ $$10.0$$ $$4.0$$ $$5.0$$ $$8.0$$ $$2.5$$
$$y =\dfrac{20}{x}$$
##### 30.
 $$x$$ $$y$$ $$1.5$$ $$320.0$$ $$2.5$$ $$115.2$$ $$4.0$$ $$45.0$$ $$6.0$$ $$20.0$$
$$y =\dfrac{720}{x^2}$$

#### 31.

The distance s a pebble falls through a thick liquid varies directly with the square of the length of time $$t$$ it falls.
1. If the pebble falls 28 centimeters in 4 seconds, express the distance it will fall as a function of time.
2. Find the distance the pebble will fall in $$6$$ seconds.
1. $$\displaystyle d = 1.75t^2$$
2. 63 cm

#### 32.

The volume, $$V\text{,}$$ of a gas varies directly with the temperature, $$T\text{,}$$ and inversely with the pressure, $$P\text{,}$$ of the gas.
1. If $$V = 40$$ when $$T = 300$$ and $$P = 30\text{,}$$ express the volume of the gas as a function of the temperature and pressure of the gas.
2. Find the volume when $$T = 320$$ and $$P = 40\text{.}$$
1. $$\displaystyle V=\dfrac{4T}{P}$$
2. $$\displaystyle 32$$

#### 33.

The demand for bottled water is inversely proportional to the price per bottle. If Droplets can sell 600 bottles at $8 each, how many bottles can the company sell at$10 each?
$$480$$ bottles

#### 34.

The intensity of illumination from a light source varies inversely with the square of the distance from the source. If a reading lamp has an intensity of 100 lumens at a distance of 3 feet, what is its intensity 8 feet away?
14.0625 lumens

#### 35.

A person’s weight, $$w\text{,}$$ varies inversely with the square of his or her distance, $$r\text{,}$$ from the center of the Earth.
1. Express $$w$$ as a function of $$r\text{.}$$ Let $$k$$ stand for the constant of variation.
2. Make a rough graph of your function.
3. How far from the center of the Earth must Neil be in order to weigh one-third of his weight on the surface? The radius of the Earth is about 3960 miles.
1. $$\displaystyle w = \dfrac{k}{r^2}$$
2. 3. $$3960\sqrt{3}\approx 6860$$ miles

#### 36.

The period, $$T\text{,}$$ of a pendulum varies directly with the square root of its length, $$L\text{.}$$
1. Express $$T$$ as a function of $$L\text{.}$$ Let $$k$$ stand for the constant of variation.
2. Make a rough graph of your function.
3. If a certain pendulum is replaced by a new one four-fifths as long as the old one, what happens to the period?

#### Exercise Group.

For Problems 37 and 38, sketch a graph to illustrate the situations.
##### 37.
Inga runs hot water into the bathtub until it is about half full. Because the water is too hot, she lets it sit for a while before getting into the tub. After several minutes of bathing, she gets out and drains the tub. Graph the water level in the bathtub as a function of time, from the moment Inga starts filling the tub until it is drained. ##### 38.
David turns on the oven and it heats up steadily until the proper baking temperature is reached. The oven maintains that temperature during the time David bakes a pot roast. When he turns the oven off, David leaves the oven door open for a few minutes, and the temperature drops fairly rapidly during that time. After David closes the door, the temperature continues to drop, but at a much slower rate. Graph the temperature of the oven as a function of time, from the moment David first turns on the oven until shortly after David closes the door when the oven is cooling. #### Exercise Group.

For Problems 39–42, sketch a graph by hand for the function.
##### 39.
$$y$$ varies directly with $$x^2\text{.}$$ The constant of variation is $$k=0.25\text{.}$$ ##### 40.
$$y$$ varies directly with $$x\text{.}$$ The constant of variation is $$k = 1.5\text{.}$$
##### 41.
$$y$$ varies inversely with $$x\text{.}$$ The constant of variation is $$k = 2\text{.}$$ ##### 42.
$$y$$ varies inversely with $$x^2\text{.}$$ The constant of variation is $$k = 4\text{.}$$

#### Exercise Group.

In Problems 43 and 44,
1. Plot the points and sketch a smooth curve through them.
2. Use your graph to discover the equation that describes the function.
##### 43.
 $$x$$ $$g(x)$$ $$2$$ $$12$$ $$3$$ $$8$$ $$4$$ $$6$$ $$6$$ $$4$$ $$8$$ $$3$$ $$12$$ $$2$$
1. 2. $$\displaystyle g(x)=\dfrac{24}{x}$$
##### 44.
 $$x$$ $$F(x)$$ $$-2$$ $$8$$ $$-1$$ $$1$$ $$0$$ $$0$$ $$1$$ $$-1$$ $$2$$ $$-8$$ $$3$$ $$-27$$
1. 2. $$\displaystyle F(x)=-x^3$$

#### Exercise Group.

In Problems 45–50,
1. Use the graph to complete the table of values.
2. By finding a pattern in the table of values, write an equation for the graph.
##### 45. $$x$$ $$0$$ $$4$$ $$8$$ $$\hphantom{000}$$ $$16$$ $$\hphantom{000}$$ $$y$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$10$$ $$\hphantom{000}$$ $$2$$
1.  $$x$$ $$0$$ $$4$$ $$8$$ $$14$$ $$16$$ $$22$$ $$y$$ $$24$$ $$20$$ $$16$$ $$10$$ $$8$$ $$2$$
2. $$\displaystyle y = 24 - x$$
##### 46. $$x$$ $$0$$ $$4$$ $$10$$ $$\hphantom{000}$$ $$14$$ $$\hphantom{000}$$ $$y$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$18$$ $$\hphantom{000}$$ $$24$$
1.  $$x$$ $$0$$ $$4$$ $$10$$ $$12$$ $$14$$ $$16$$ $$y$$ $$0$$ $$6$$ $$15$$ $$18$$ $$21$$ $$24$$
2. $$\displaystyle y = \dfrac{3}{2}x$$
##### 47. $$x$$ $$0$$ $$\hphantom{000}$$ $$4$$ $$\hphantom{000}$$ $$16$$ $$25$$ $$y$$ $$\hphantom{000}$$ $$1$$ $$\hphantom{000}$$ $$3$$ $$\hphantom{000}$$ $$\hphantom{000}$$
1.  $$x$$ $$0$$ $$1$$ $$4$$ $$9$$ $$16$$ $$25$$ $$y$$ $$0$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$
2. $$\displaystyle y = \sqrt{x}$$
##### 48. $$x$$ $$\hphantom{000}$$ $$0.5$$ $$1$$ $$1.5$$ $$\hphantom{000}$$ $$4$$ $$y$$ $$4$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$0.5$$ $$\hphantom{000}$$
1.  $$x$$ $$0.25$$ $$0.50$$ $$1.00$$ $$1.50$$ $$2.00$$ $$4.00$$ $$y$$ $$4.00$$ $$2.00$$ $$1.00$$ $$0.67$$ $$0.50$$ $$0.25$$
2. $$\displaystyle y = \dfrac{1}{x}$$
##### 49. $$x$$ $$-3$$ $$-2$$ $$\hphantom{000}$$ $$0$$ $$1$$ $$2$$ $$y$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$-3$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$\hphantom{000}$$
1.  $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$2$$ $$y$$ $$5$$ $$0$$ $$-3$$ $$-4$$ $$-3$$ $$0$$
2. $$\displaystyle y = x^2-4$$ $$x$$ $$-3$$ $$-2$$ $$\hphantom{000}$$ $$0$$ $$1$$ $$\hphantom{000}$$ $$y$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$8$$ $$\hphantom{000}$$ $$\hphantom{000}$$ $$-7$$
1.  $$x$$ $$-3$$ $$-2$$ $$-1$$ $$0$$ $$1$$ $$4$$ $$y$$ $$0$$ $$5$$ $$8$$ $$9$$ $$8$$ $$-7$$
2. $$\displaystyle y = 9 - x^2$$