## Section 5.7 Chapter 5 Summary and Review

### Subsection 5.7.1 Glossary

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

### Subsection 5.7.2 Key Concepts

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

A graph represents a function if and only if every vertical line intersects the graph in at most one point.- We can use a graphical technique to solve equations and inequalities.
- \(b\) is the cube root of \(a\) if \(b\) cubed equals \(a\text{.}\) In symbols, we write\begin{equation*} \blert{b=\sqrt[3]{a}~~~~\text{if}~~~~b^3=a} \end{equation*}
#### 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*}- 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.
- The maximum or minimum of a quadratic function occurs at the vertex.
- 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[3]{x}~~~~~~f(x)=\dfrac{1}{x}~~~~~~f(x)=\dfrac{1}{x^2} \end{gather*}
- Two variables are directly proportional if the ratios of their corresponding values are always equal.
#### 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.- 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.
- Direct variation has the following scaling property: increasing \(x\) by any factor causes \(y\) to increase by the same factor.
#### 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.- If the ratio \(\dfrac{y}{x^n}\) is constant, then \(y\) varies directly with \(x^n\text{.}\)
#### 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.#### 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.- If the product \(~yx^n~\) is constant and \(n\) is positive, then \(y\) varies inversely with \(x^n\text{.}\)
- A graph that bends upward is called concave up, and one that bends downward is concave down.

### Exercises 5.7.3 Chapter 5 Review Problems

#### Exercise Group.

Which of the tables in Problems 1–4 describe functions? Why or why not?

##### 1.

\(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |

\(y\) | \(6\) | \(0\) | \(1\) | \(2\) | \(6\) | \(8\) |

## Answer.

A function: Each \(x\) has exactly one associated \(y\)-value.

##### 2.

\(p\) | \(3\) | \(-3\) | \(2\) | \(-2\) | \(-2\) | \(0\) |

\(q\) | \(2\) | \(-1\) | \(4\) | \(-4\) | \(3\) | \(0\) |

## Answer.

Not a function

##### 3.

Student | Score on IQ test |
Score on SAT test |

(A) | \(118\) | \(649\) |

(B) | \(98\) | \(450\) |

(C) | \(110\) | \(590\) |

(D) | \(105\) | \(520\) |

(E) | \(98\) | \(490\) |

(F) | \(122\) | \(680\) |

## Answer.

Not a function: The IQ of \(98\) has two possible SAT scores.

##### 4.

Student | Correct answers on math quiz |
Quiz grade |

(A) | \(13\) | \(85\) |

(B) | \(15\) | \(89\) |

(C) | \(10\) | \(79\) |

(D) | \(12\) | \(82\) |

(E) | \(16\) | \(91\) |

(F) | \(18\) | \(95\) |

## Answer.

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.

## Answer.

\(N(10) = 7000\text{:}\) 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.

## Answer.

\(H(16) = 3\text{:}\) 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 \(~~F(-3)\)

## Answer.

\(F(0) = 1, ~~F(-3) =\sqrt{37}\)

##### 8.

\(G(x)=\sqrt[3]{x-8}\text{,}\) \(~~G(0)~~\) and \(~~G(20)\)

## Answer.

\(G(0) = -2, ~~G(20) =\sqrt[3]{12}\)

##### 9.

\(h(v)=6-\abs{4-2v} \text{,}\) \(~~h(8)~~\) and \(~~h(-8)\)

## Answer.

\(h(8) = -6, ~~h(-8) = -14\)

##### 10.

\(m(p)=\dfrac{120}{p+15} \text{,}\) \(~~m(5)~~\) and \(~~m(-40)\)

## Answer.

\(m(5) = 6, ~~m(-40) =-4.8\)

#### 11.

\(P(x)=x^2-6x+5\)

- Compute \(P(0)\text{.}\)
- Find all values of \(x\) for which \(P(x)=0\text{.}\)

## Answer.

- \(\displaystyle P(0)=5\)
- x=5,~x=1

#### 12.

\(R(x)=\sqrt{4-x^2}\)

- Compute \(R(0)\text{.}\)
- Find all values of \(x\) for which \(R(x)=0\text{.}\)

## Answer.

- R(0)=2
- \(\displaystyle x=2,~x=-2\)

#### Exercise Group.

For Problems 13 and 14, refer to the graphs to answer the questions.

##### 13.

- Find \(f (-2)\) and \(f (2)\text{.}\)
- For what value(s) of \(t\) is \(f (t) = 4\text{?}\)
- Find the \(t\)- and \(f(t)\)-intercepts of the graph.
- What is the maximum value of \(f\text{?}\) For what value(s) of \(t\) does \(f\) take on its maximum value?

## Answer.

- \(\displaystyle f (-2) = 3, ~~f (2) = 5\)
- \(\displaystyle t = 1, ~~t = 3\)
- \(t\)-intercepts \((-3, 0), (4, 0)\text{;}\) \(f (t)\)-intercept: \((0, 2)\)
- Maximum value of \(5\) occurs at \(t = 2\)

##### 14.

- Find \(P(-3)\) and \(P(3)\text{.}\)
- For what value(s) of \(z\) is \(P(z) = 2\text{?}\)
- Find the \(z\)- and \(P(z)\)-intercepts of the graph.
- What is the minimum value of \(P\text{?}\) For what value(s) of \(z\) does \(P\) take on its minimum value?

## Answer.

- \(\displaystyle P(-3)=-2, ~~P(3)=3\)
- \(\displaystyle z = -5,~ \dfrac{-1}{2},~4\)
- \((-4, 0), (-1, 0), (5,0)\text{;}\) \((0, 3)\)
- Maximum value of \(-3\) occurs at \(z = -2\)

#### Exercise Group.

Which of the graphs in Problems 15–18 represent functions?

##### 15.

## Answer.

Function

##### 16.

## Answer.

not a function

##### 17.

## Answer.

Not a function

##### 18.

## Answer.

Function

#### Exercise Group.

For Problems 19–22, graph the function by hand.

##### 19.

\(f(t)=-2t+4\)

## Answer.

##### 20.

\(g(s)=\dfrac{-2}{3}s-2\)

##### 21.

\(p(x)=9-x^2\)

## Answer.

##### 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.

\(y=\sqrt[3]{x} \)

- Solve \(\sqrt[3]{x} = 0.8\)
- Solve \(\sqrt[3]{x} = 1.5\)
- Solve \(\sqrt[3]{x}\gt 1.7 \)
- Solve \(\sqrt[3]{x}\le 1.26 \)

## Answer.

- \(\displaystyle x = \dfrac{1}{2}= 0.5\)
- \(\displaystyle x = \dfrac{27}{8}\approx 3.4\)
- \(\displaystyle x \gt 4.9\)
- \(\displaystyle x \le 2.0\)

##### 24.

\(y=\dfrac{1}{x} \)

- Solve \(\dfrac{1}{x} = 2.5\)
- Solve \(\dfrac{1}{x} = 0.3125\)
- Solve \(\dfrac{1}{x}\ge 0.\overline{2} \)
- Solve \(\dfrac{1}{x}\lt 5\)

## Answer.

- \(\displaystyle x = 0.4\)
- \(\displaystyle x = 3.2\)
- \(\displaystyle 0 \lt x \le 4.5\)
- \(x \lt 0\) or \(x \gt 0.2\)

##### 25.

\(y=\dfrac{1}{x^2} \)

- Solve \(\dfrac{1}{x^2} = 0.03\)
- Solve \(\dfrac{1}{x^2} = 6.25\)
- Solve \(\dfrac{1}{x^2}\gt 0.16 \)
- Solve \(\dfrac{1}{x^2}\le 4\)

## Answer.

- \(\displaystyle x\approx\pm 5.8 \)
- \(\displaystyle x = \pm 0.4\)
- \(-2.5\lt x \lt 0\) or \(0\lt x\lt 2.5\)
- \(x \le -0.5\) or \(x\ge 0.5\)

##### 26.

\(y=\sqrt{x} \)

- Solve \(\sqrt{x} = 0.707\)
- Solve \(\sqrt{x} = 1.7\)
- Solve \(\sqrt{x}\lt 1.5 \)
- Solve \(\sqrt{x}\ge 1.3 \)

## Answer.

- \(\displaystyle x = 0.5\)
- \(\displaystyle x = 2.9\)
- \(\displaystyle 0 \le x \lt 2.3\)
- \(\displaystyle x \ge 1.7\)

#### 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.

\(x\) | \(y\) |

\(2\) | \(4.8\) |

\(5\) | \(30.0\) |

\(8\) | \(76.8\) |

\(11\) | \(145.2\) |

## Answer.

\(y = 1.2x^2\)

##### 28.

\(x\) | \(y\) |

\(1.4\) | \(75.6\) |

\(2.3\) | \(124.2\) |

\(5.9\) | \(318.6\) |

\(8.3\) | \(448.2\) |

## Answer.

\(y = 54x\)

##### 29.

\(x\) | \(y\) |

\(0.5\) | \(40.0\) |

\(2.0\) | \(10.0\) |

\(4.0\) | \(5.0\) |

\(8.0\) | \(2.5\) |

## Answer.

\(y =\dfrac{20}{x} \)

##### 30.

\(x\) | \(y\) |

\(1.5\) | \(320.0\) |

\(2.5\) | \(115.2\) |

\(4.0\) | \(45.0\) |

\(6.0\) | \(20.0\) |

## Answer.

\(y =\dfrac{720}{x^2} \)

#### 31.

The distance s a pebble falls through a thick liquid varies directly with the square of the length of time \(t\) it falls.

- If the pebble falls 28 centimeters in 4 seconds, express the distance it will fall as a function of time.
- Find the distance the pebble will fall in \(6\) seconds.

## Answer.

- \(\displaystyle d = 1.75t^2\)
- 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.

- 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.
- Find the volume when \(T = 320\) and \(P = 40\text{.}\)

## Answer.

- \(\displaystyle V=\dfrac{4T}{P}\)
- \(\displaystyle 32\)

#### 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?

## Answer.

\(480\) 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?

## Answer.

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.

- Express \(w\) as a function of \(r\text{.}\) Let \(k\) stand for the constant of variation.
- Make a rough graph of your function.
- 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.

## Answer.

- \(\displaystyle w = \dfrac{k}{r^2}\)
- \(3960\sqrt{3}\approx 6860\) miles

#### 36.

The period, \(T\text{,}\) of a pendulum varies directly with the square root of its length, \(L\text{.}\)

- Express \(T\) as a function of \(L\text{.}\) Let \(k\) stand for the constant of variation.
- Make a rough graph of your function.
- 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.

## Answer.

##### 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.

## Answer.

#### 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 \(k=0.25\text{.}\)

## Answer.

##### 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 \(k = 2\text{.}\)

## Answer.

##### 42.

\(y\) varies inversely with \(x^2\text{.}\) The constant of variation is \(k = 4\text{.}\)

#### Exercise Group.

In Problems 43 and 44,

- Plot the points and sketch a smooth curve through them.
- Use your graph to discover the equation that describes the function.

##### 43.

\(x\) | \(g(x)\) |

\(2\) | \(12\) |

\(3\) | \(8\) |

\(4\) | \(6\) |

\(6\) | \(4\) |

\(8\) | \(3\) |

\(12\) | \(2\) |

## Answer.

- \(\displaystyle g(x)=\dfrac{24}{x} \)

##### 44.

\(x\) | \(F(x)\) |

\(-2\) | \(8\) |

\(-1\) | \(1\) |

\(0\) | \(0\) |

\(1\) | \(-1\) |

\(2\) | \(-8\) |

\(3\) | \(-27\) |

## Answer.

- \(\displaystyle F(x)=-x^3 \)

#### Exercise Group.

In Problems 45–50,

- Use the graph to complete the table of values.
- By finding a pattern in the table of values, write an equation for the graph.

##### 45.

\(x\) | \(0\) | \(4\) | \(8\) | \(\hphantom{000}\) | \(16\) | \(\hphantom{000}\) |

\(y\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(10\) | \(\hphantom{000}\) | \(2\) |

## Answer.

\(x\) \(0\) \(4\) \(8\) \(14\) \(16\) \(22\) \(y\) \(24\) \(20\) \(16\) \(10\) \(8\) \(2\) - \(\displaystyle y = 24 - x\)

##### 46.

\(x\) | \(0\) | \(4\) | \(10\) | \(\hphantom{000}\) | \(14\) | \(\hphantom{000}\) |

\(y\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(18\) | \(\hphantom{000}\) | \(24\) |

## Answer.

\(x\) \(0\) \(4\) \(10\) \(12\) \(14\) \(16\) \(y\) \(0\) \(6\) \(15\) \(18\) \(21\) \(24\) - \(\displaystyle y = \dfrac{3}{2}x\)

##### 47.

\(x\) | \(0\) | \(\hphantom{000}\) | \(4\) | \(\hphantom{000}\) | \(16\) | \(25\) |

\(y\) | \(\hphantom{000}\) | \(1\) | \(\hphantom{000}\) | \(3\) | \(\hphantom{000}\) | \(\hphantom{000}\) |

## Answer.

\(x\) \(0\) \(1\) \(4\) \(9\) \(16\) \(25\) \(y\) \(0\) \(1\) \(2\) \(3\) \(4\) \(5\) - \(\displaystyle y = \sqrt{x} \)

##### 48.

\(x\) | \(\hphantom{000}\) | \(0.5\) | \(1\) | \(1.5\) | \(\hphantom{000}\) | \(4\) |

\(y\) | \(4\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(0.5\) | \(\hphantom{000}\) |

## Answer.

\(x\) \(0.25\) \(0.50\) \(1.00\) \(1.50\) \(2.00\) \(4.00\) \(y\) \(4.00\) \(2.00\) \(1.00\) \(0.67\) \(0.50\) \(0.25\) - \(\displaystyle y = \dfrac{1}{x}\)

##### 49.

\(x\) | \(-3\) | \(-2\) | \(\hphantom{000}\) | \(0\) | \(1\) | \(2\) |

\(y\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(-3\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(\hphantom{000}\) |

## Answer.

\(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(y\) \(5\) \(0\) \(-3\) \(-4\) \(-3\) \(0\) - \(\displaystyle y = x^2-4 \)

##### 50.

\(x\) | \(-3\) | \(-2\) | \(\hphantom{000}\) | \(0\) | \(1\) | \(\hphantom{000}\) |

\(y\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(8\) | \(\hphantom{000}\) | \(\hphantom{000}\) | \(-7\) |

## Answer.

\(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(4\) \(y\) \(0\) \(5\) \(8\) \(9\) \(8\) \(-7\) - \(\displaystyle y = 9 - x^2\)

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