Chapter 2

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of the linear function \(5 x+6 y-30-0\) is a line passing through the point \((6,0)\) with slope \(-\frac{5}{6}\).

Use a graphing utility to graph each function. Use a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$h(x)=2-x^{\frac{2}{5}}$$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)-\frac{1}{2} \sqrt[3]{x-2} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I must have made a mistake in finding the composite functions \(f \circ g\) and \(g \circ f,\) because I notice that \(f \circ g\) is not the same function as \(g \circ f\)

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ r(x)- \frac 1 2 \sqrt[3]{x+2}-2 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph is decreasing on \((-\infty, a)\) and increasing on \((a, \infty)\) so \(f(a)\) must be a relative maximum.

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ r(x)-\frac{1}{2} \sqrt[3]{x-2}+2 $$

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)--\sqrt[3]{x+2} $$

Prove that the equation of a line passing through \((a, 0)\) and \((0, b)(a \neq 0, b \neq 0)\) can be written in the form \(\frac{x}{a}+\frac{y}{b}-1\) Why is this called the intercept form of a line?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed $$f(x)=\left\\{\begin{array}{lll}2 & \text { if } & x \neq 4 \\\3 & \text { if } & x=4\end{array}\right.$$ and one piece of my graph is a single point.

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ h(x)--\sqrt[3]{x-2} $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I noticed that the difference quotient is always zero if \(f(x)=c,\) where \(c\) is any constant.

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{-x-2} $$

We used the data in a bar graph to develop linear functions that modeled the percentage of never-married American females and males, ages \(25-29\) For this group exercise, you might find it helpful to pattern your work after Exercises 87 and \(88 .\) Group members should begin by consulting an almanac, newspaper, magazine, or the Internet to find data that appear to lie approximately on or near a line. Working by hand or using a graphing utility, group members should construct scatter plots for the data that were assembled. If working by hand, draw a line that approximately fits the data in each scatter plot and then write its equation as a function in slope-intercept form. If using a graphing utility, obtain the equation of each regression line. Then use each linear function's equation to make predictions about what might occur in the future. Are there circumstances that might affect the accuracy of the prediction? List some of these circumstances.

Sketch the graph of \(f\) using the following properties. (More than one correct graph is possible.) \(f\) is a piecewise function that is decreasing on \((-\infty, 2), f(2)=0, f\) is increasing on \((2, \infty),\) and the range of \(f\) is \([0, \infty)\)

Begin by graphing the cube root function, \(f(x)-\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$ g(x)-\sqrt[3]{-x+2} $$

Write the slope-intercept form of the equation of the line passing through \((-3,1)\) whose slope is the same as the line whose equation is \(y-2 x+1\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Prove that if \(f\) and \(g\) are even functions, then \(f g\) is also an even function.

Define a piecewise function on the intervals \((-\infty, 2],(2,5)\) and \([5, \infty)\) that does not "jump" at 2 or 5 such that one piece is a constant function, another piece is an increasing function, and the third piece is a decreasing function.

Write an equation in general form of the line passing through \((3,-5)\) whose slope is the negative reciprocal (the reciprocal with the opposite sign) of \(-\frac{1}{4}\)

Suppose that \(h(x)=\frac{f(x)}{g(x)} .\) The function \(f\) can be even, odd, or neither. The same is true for the function \(g .\) a. Under what conditions is \(h\) definitely an even function? b. Under what conditions is \(h\) definitely an odd function?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Consider the function defined by $$ [(-2,4),(-1,1),(1,1),(2,4)] $$ Reverse the components of each ordered pair and write the resulting relation. Is this relation a function?

Group members who have cellphone plans should describe the total monthly cost of the plan as follows: ______ per month buys _______minutes. Additional time costs $________ per minute. (For simplicity, ignore other charges.) The group should select any three plans, from "basic" to "premier." For each plan selected, write a piecewise function that describes the plan and graph the function. Graph the three functions in the same rectangular coordinate system. Now examine the graphs. For any given number of calling minutes, the best plan is the one whose graph is lowest at that point. Compare the three calling plans. Is one plan always a better deal than the other two? If not, determine the interval of calling minutes for which each plan is the best deal.

Will help you prepare for the material covered in the next section. $$ \text { Solve for } y: \quad x-\frac{5}{y}+4 $$

Will help you prepare for the material covered in the next section. $$ \text { Solve for } y: \quad x-y^{2}-1, y \geq 0 $$

Find the ordered pairs (______, 0) and (0,______) satisfying 4 x-3 y-6=0

$$\text { Solve for } y: 3 x+2 y-4=0$$

What must be done to a function's equation so that its graph is shifted vertically upward?

What must be done to a function's equation so that its graph is shifted horizontally to the right?

What must be done to a function's equation so that its graph is reflected about the \(x\) -axis?

What must be done to a function's equation so that its graph is reflected about the \(y\) -axis?

What must be done to a function's equation so that its graph is stretched vertically?

What must be done to a function's equation so that its graph is shrunk horizontally?

a. Use a graphing utility to graph \(f(x)=x^{2}+1\) b. Graph \(f(x)=x^{2}+1, g(x)=f(2 x), h(x)=f(3 x),\) and \(k(x)=f(4 x)\) in the same viewing rectangle. c. Describe the relationship among the graphs of \(f, g, h\) and \(k,\) with emphasis on different values of \(x\) for points on all four graphs that give the same \(y\) -coordinate. d. Generalize by describing the relationship between the graph of \(f\) and the graph of \(g,\) where \(g(x)=f(c x)\) for \(c>1\) e. Try out your generalization by sketching the graphs of \(f(c x)\) for \(c=1, c=2, c=3,\) and \(c=4\) for a function of your choice.

Make Sense? During the winter, you program your home thermostat so that at midnight, the temperature is \(55^{\circ} .\) This temperature is maintained until 6 a.m Then the house begins to warm up so that by 9 a.m the temperature is \(65^{\circ} .\) At 6 p.m the house begins to cool. By 9 p.m the temperature is again \(55^{\circ}\). The graph illustrates home temperature, \(f(t),\) as a function of hours after midnight, t. (Graph can't copy) Determine whether each statement makes sense or does not make sense, and explain your reasoning. If the statement makes sense, graph the new function on the domain \([0,24]\). If the statement does not make sense, correct the function in the statement and graph the corrected function on the domain \([0,24]\) I decided to keep the house \(5^{\circ}\) cooler than before, so I reprogrammed the thermostat to \(y=f(t)-5\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=|x|\) and \(g(x)=|x+3|+3,\) then the graph of \(g\) is a translation of the graph of \(f\) three units to the right and three units upward.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=x^{3}\) and \(g(x)=-(x-3)^{3}-4,\) then the graph of \(g\) can be obtained from the graph of \(f\) by moving \(f\) three units to the right, reflecting about the \(x\) -axis, and then moving the resulting graph down four units.

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$ y-f(x-3) $$

Perform the indicated operation or operations. $$ (2 x-1)\left(x^{2}+x-2\right) $$

Perform the indicated operation or operations. $$ (f(x))^{2}-2 f(x)+6, \text { where } f(x)-3 x-4 $$

Perform the indicated operation or operations. Simplify: $$ \frac{2}{\frac{3}{x}-1} $$