Chapter 4

When on vacation, the Ross family always travels the same number of miles each day. a. Does the time that they travel each day vary inversely as the rate at which they travel? b. On the first day the Ross family travels for 3 hours at an average rate of 60 miles per hour and on the second day they travel for 4 hours. What was their average rate of speed on the second day?

In \(12-17,\) use a graph to find the solution set of each inequality. $$ x^{2}-2 x+1 < 0 $$

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=-x\)

a. Sketch the graph of \(y=|x| .\) b. Sketch the graph of \(y=-|x| .\) c. Describe the graph of \(y=-|x|\) in terms of the graph of \(y=|x|\)

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \left\\{(x, y) : y=-x^{2}+3 x-2\right\\} $$

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \left\\{(x, y) : y=x^{2}+2 \text { for } 0 \leq x \leq 5\right\\} $$

Create your own function \(\mathrm{f}(x)\) and show that \(\mathrm{f}(x)=2 \mathrm{f}(x)\) . Explain why this result is true in general.

Megan traveled 165 miles to visit friends. On the return trip she was delayed by construction and had to reduce her average speed by 22 miles per hour. The return trip took 2 hours longer. What was the time and average speed for each part of the trip?

In \(11-18 :\) a. Find \(h(x)\) when \(h(x)=g(f(x)) .\) b. What is the domain of \(h(x) ?\) c. What is the range of \(\mathrm{h}(x) ?\) d. Graph \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=|2+x|, \mathrm{g}(x)=-x $$

In \(12-17,\) use a graph to find the solution set of each inequality. $$ -x^{2}+6 x-5 < 0 $$

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=\frac{8}{x}\)

a. Sketch the graph of \(y=|x| .\) b. Sketch the graph of \(y=2|x|\) c. Sketch the graph of \(y=\frac{1}{2}|x|\) d. Describe the graph of \(y=a|x|\) in terms of the graph of \(y=|x|\)

The sales tax \(t\) on a purchase is a function of the amount \(a\) of the purchase. The sales tax rate in the city of East chester is 8\(\% .\) a. Write a rule in function notation that can be used to determine the sales tax on a pur- chase in East chester. b. What is a reasonable domain for this function? c. Find the sales tax when the purchase is \(\$ 5.00 .\) d. Find the sales tax when the purchase is \(\$ 16.50 .\)

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \\{(x, y) : y=\sqrt{2 x}\\} $$

In \(17-20 :\) a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers. $$ f(x)=4 x-3 $$

Ian often buys in large quantities. A few months ago he bought several cans of frozen orange juice for \(\$ 24 .\) The next time lan purchased frozen orange juice, the price had increased by \(\$ 0.10\) per can and he bought 1 less can for the same total price. What was the price per can and the numbers of cans purchased each time?

In \(11-18 :\) a. Find \(h(x)\) when \(h(x)=g(f(x)) .\) b. What is the domain of \(h(x) ?\) c. What is the range of \(\mathrm{h}(x) ?\) d. Graph \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=5-x, \mathrm{g}(x)=|x| $$

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=\frac{1}{2} x\)

a. Draw the graphs of \(y=|x+3|\) and \(y=5\) b. From the graph drawn in a, determine the solution set of \(|x+3|=5 .\) C. From the graph drawn in a, determine the solution set of \(|x+3| > 5\) d. From the graph drawn in a, determine the solution set of \(|x+3| < 5\)

A muffin shop's weckly profit is a function of the number of muffins \(m\) that it sells. The equation approximating the weekly profit is \(f(m)=0.60 m-900 .\) a. Draw the graph showing the relationship between the number of muffins sold and the profit. b. What is the profit if \(2,000\) muffins are sold? c. How many muffins must be sold for the shop to make a profit of \(\$ 900 ?\)

In \(17-20 :\) a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers. $$ g(x)=x-5 $$

In \(11-18 :\) a. Find \(h(x)\) when \(h(x)=g(f(x)) .\) b. What is the domain of \(h(x) ?\) c. What is the range of \(\mathrm{h}(x) ?\) d. Graph \(\mathrm{h}(x)\) $$ \mathrm{f}(x)=3 x-1, \mathrm{g}(x)=\frac{1}{3}(x+1) $$

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(y=\sqrt{x}\)

a. Draw the graphs of \(y=-|x-4|\) and \(y=-2\) b. From the graph drawn in a, determine the solution set of \(-|x-4|=-2\) c. From the graph drawn in a, determine the solution set of \(-|x-4| > -2\) d. From the graph drawn in a, determine the solution set of \(-|x-4| < -2\)

In \(17-20 :\) a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers. $$ f(x)=\frac{x+5}{3} $$

The sum of the lengths of the legs of a right triangle is 20 feet. a. If \(x\) is the measure of one of the legs, represent the measure of the other leg in terms of \(x .\) b. If \(y\) is the area of the triangle, express the area in terms of \(x .\) c. Draw the graph of the function written in \(\mathbf{b}\) . d. What are the dimensions of the triangle with the largest area?

In \(13-20\) : a. Graph each function. b. Is the function a direct variation? \(c\) . Is the function one-to-one? \(\frac{y}{x}=2\)

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \\{(x, y) : y=\sqrt{3-x}\\} $$

In \(17-20 :\) a. Find the inverse of each given function. b. Describe the domain and range of each given function and its inverse in terms of the largest possible subset of the real numbers. $$ \mathrm{f}(x)=\sqrt{x} $$

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}-25=0 $$

A polynomial function of degree three, \(\mathrm{p}(x),\) intersects the \(x\) -axis at \((-4,0),(-2,0),\) and \((3,0)\) and intersects the \(y\) -axis at \((0,-24) .\) Find \(\mathrm{p}(x) .\)

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \left\\{(x, y) : y=\frac{1}{\sqrt{x+1}}\right\\} $$

If \(\mathrm{f}=\\{(x, y) : y=5 x\\}\) is a direct variation function, find \(\mathrm{f}^{-1}\)

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}-2 x-2 y-7=0 $$

\(\operatorname{In} 19-22,\) let \(\mathrm{f}(x)=|x| \cdot\) Find \(\mathrm{f}(\mathrm{g}(x))\) and \(\mathrm{g}(\mathrm{f}(x))\) for each given function. $$ g(x)=2 x+3 $$

a. Sketch the graph of \(y=x^{2} .\) b. Sketch the graph of \(y=x^{2}+2\) c. Sketch the graph of \(y=x^{2}-3\) d. Describe the graph of \(y=x^{2}+a\) in terms of the graph of \(y=x^{2}\) . e. What transformation maps \(y=x^{2}\) to \(y=x^{2}+a ?\)

Is every linear function a direct variation?

In \(12-23,\) each set is a function from set \(A\) to set \(B .\) a. What is the largest subset of the real numbers that can be set \(A\) , the domain of the given function? b. If set \(A=\operatorname{set} B,\) is the function onto? Justify your answer. $$ \left\\{(x, y) : y=\frac{1}{x^{2}+1}\right\\} $$

If \(\mathrm{g}=\\{(x, y) : y=7-x\\},\) find \(\mathrm{g}^{-1}\) if it exists. Is it possible for a function to be its own inverse?

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}+2 x-4 y+1=0 $$

a. Sketch the graph of \(y=x^{2}\) . b. Sketch the graph of \(y=(x+2)^{2}\) c. Sketch the graph of \(y=(x-3)^{2}\) d. Describe the graph of \(y=(x+a)^{2}\) in terms of the graph of \(y=x^{2}\) e. What transformation maps \(y=x^{2}\) to \(y=(x+a)^{2} ?\)

Is the direct variation of two variables always a linear function?

Does \(y=x^{2}\) have an inverse function if the domain is the set of real numbers? Justify your answer.

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}-6 x+2 y-6=0 $$

a. Sketch the graph of \(y=x^{2}\) b. Sketch the graph of \(y=-x^{2}\) c. Describe the graph of \(y=-x^{2}\) in terms of the graph of \(y=x^{2}\) . d. What transformation maps \(y=x^{2}\) to \(y=-x^{2} ?\)

In \(23-28,\) write an equation of the direct variation described. The cost of tickets, \(c,\) is directly proportional to the number of tickets purchased, \(n .\) One ticket costs six dollars.

In \(20-27\) : a. Write each equation in center-radius form. b. Find the coordinates of the center. . Find the radius of the circle. $$ x^{2}+y^{2}+6 x-6 y+6=0 $$

If \(\mathrm{p}(x)=2\) and \(\mathrm{q}(x)=x+2,\) find \(\mathrm{p}(\mathrm{q}(5))\) and \(\mathrm{q}(\mathrm{p}(5))\)

a. Sketch the graph of \(y=x^{2}\) b. Sketch the graph of \(y=3 x^{2}\) c. Sketch the graph of \(y=\frac{1}{3} x^{2}\) d. Describe the graph of \(y=a x^{2}\) in terms of the graph of \(y=x^{2}\) when \(a > 1\) e. Describe the graph of \(y=a x^{2}\) in terms of the graph of \(y=x^{2}\) when \(0 < a < 1\)

In \(23-28,\) write an equation of the direct variation described. The distance in miles, \(d\) , that Mr. Spencer travels is directly proportional to the length of time in hours, \(t,\) that he travels at 35 miles per hour.