Pepperoni Pizzas For a statistics project, a student

In Exercises 1-10, evaluate the expression without using a calculator. \(|-59.34|\)

In Exercises 1-10, evaluate the expression without using a calculator. \(|5-\pi|\)

In Exercises 1-10, evaluate the expression without using a calculator. \(|\pi-7|\)

In Exercises 1-10, evaluate the expression without using a calculator. \(\sqrt{(-3)^{2}}\)

In Exercises 1-10, evaluate the expression without using a calculator. ln(1)

In Exercises 1-10, evaluate the expression without using a calculator. e0

In Exercises 1-10, evaluate the expression without using a calculator. \((\sqrt[3]{3})^{3}\)

In Exercises 1-10, evaluate the expression without using a calculator. \(\sqrt[3]{(-15)^{3}}\)

In Exercises 1-10, evaluate the expression without using a calculator. \(\sqrt[3]{-8^{2}}\)

In Exercises 1-10, evaluate the expression without using a calculator. \(|1-\pi|-\pi\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 112, each graph is a slight variation on the graph of one of the twelve basic functions described in this section. Match the graph to one of the twelve functions (a)(l) and then support your answer by checking the graph on your calculator. (All graphs are shown in the window [-4.7, 4.7] by [-3.1, 3.1].) (a) \(y=-\sin x\) (b)\(y=\cos x+1\) (c) \(y=e^{x}-2\) (d) \(y=(x+2)^{3}\) (e) \(y=x^{3}+1\) (f) \(y=(x-1)^{2}\) (g) \(y=|x|-2\) (h) \(y=-1 / x\) (i) \(y=-x\) (j) \(y=-\sqrt{x}\) (k) \(y=\operatorname{int}(x+1)\) (l) \(y=2-4 /\left(1+e^{-x}\right)\)

In Exercises 13-18, identify which of Exercises 1-12 display functions that fit the description given. The function whose domain excludes zero

In Exercises 13-18, identify which of Exercises 1-12 display functions that fit the description given. The function whose domain consists of all nonnegative real numbers

In Exercises 13-18, identify which of Exercises 1-12 display functions that fit the description given. The two functions that have at least one point of discontinuity

In Exercises 13-18, identify which of Exercises 1-12 display functions that fit the description given. The function that is not a continuous function

In Exercises 13-18, identify which of Exercises 1-12 display functions that fit the description given. The six functions that are bounded below

In Exercises 13-18, identify which of Exercises 1-12 display functions that fit the description given. The four functions that are bounded above

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The four functions that are odd

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The six functions that are increasing on their entire domains

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The three functions that are decreasing on the interval (- \(\infty\), 0)

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The three functions with infinitely many local extrema

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The three functions with no zeros

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The three functions with range {all real numbers}

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The four functions that do not have end behavior \(\lim _{x \rightarrow+\infty} f(x)=+\infty\)

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The three functions with end behavior \(\lim _{x \rightarrow-\infty} f(x)=-\infty\)

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The four functions whose graphs look the same when turned upside-down and flipped about the y-axis

In Exercises 19-28, identify which of the twelve basic functions fit the description given. The two functions whose graphs are identical except for a horizontal shift

In Exercises 29-34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. \(f(x)=x^{2}-5\)

In Exercises 29-34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. \(g(x)=|x-4|\)

In Exercises 29-34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. \(h(x)=\ln (x+6)\)

In Exercises 29-34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. \(k(x)=1 / x+3\)

In Exercises 29-34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. \(s(x)=\operatorname{int}(x / 2)\)

In Exercises 29-34, use your graphing calculator to produce a graph of the function. Then determine the domain and range of the function by looking at its graph. \(p(x)=(x+3)^{2}\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(r(x)=\sqrt{ } x-10\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(f(x)=\sin (x)+5\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(f(x)=3 /\left(1+e^{-x}\right)\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \9q(x)=e^{x}+2\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(h(x)=|x|-10\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(g(x)=4 \cos (x)\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(s(x)=|x-2|\)

In Exercises 35-42, graph the function. Then answer the following questions: (a) On what interval, if any, is the function increasing? Decreasing? (b) Is the function odd, even, or neither? (c) Give the functions extrema, if any. (d) How does the graph relate to a graph of one of the twelve basic functions? \(f(x)=5-\operatorname{abs}(x)\)

Find the horizontal asymptotes for the graph shown in Exercise 11.

Find the horizontal asymptotes for the graph of ƒ(x) in Exercise 37.

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(f(x)=\left\{\begin{array}{ll}x & \text { if } x \leq 0 \\x^{2} & \text { if } x>0\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(g(x)=\left\{\begin{array}{ll}x^{3} & \text { if } x \leq 0 \\e^{x} & \text { if } x>0\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(h(x)=\left\{\begin{array}{ll}|x| & \text { if } x<0 \\\sin x & \text { if } x \geq 0\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(w(x)=\left\{\begin{array}{ll}1 / x & \text { if } x<0 \\\sqrt{x} & \text { if } x \geq 0\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(f(x)=\left\{\begin{array}{ll}\cos x & \text { if } x \leq 0 \\e^{x} & \text { if } x>0\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(g(x)=\left\{\begin{array}{ll}|x| & \text { if } x<0 \\x^{2} & \text { if } x \geq 0\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(f(x)=\left\{\begin{array}{ll}-3-x & \text { if } x \leq 0 \\1 & \text { if } 0<x<1 \\x^{2} & \text { if } x \geq 1\end{array}\right.\)

In Exercises 45-52, sketch the graph of the piecewise-defined function. (Try doing it without a calculator.) In each case, give any points of discontinuity. \(f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x<-1 \\|x| & \text { if }-1 \leq x<1 \\\text { int }(x) & \text { if } x \geq 1\end{array}\right.\)

The function \(f(x)=\sqrt{x^{2}}\) is one of our twelve basic functions written in another form. (a) Graph the function and identify which basic function it is. (b) Explain algebraically why the two functions are equal.

The function \(g(x)=\sqrt{x^{2}+0.0001}-0.01\) is not one of our twelve basic functions written in another form. (a) Graph the function and identify which basic function it appears to be. (b) Verify numerically that it is not the basic function that it appears to be.

The function f(x) = ln(ex) is one of our twelve basic functions written in another form. (a) Graph the function and identify which basic function it is. (b) Explain how the equivalence of the two functions in (a) shows that the natural logarithm function is not bounded above (even though it appears to be bounded above in Figure 1.42).

Let f(x) be the function that gives the cost, in cents, to mail a first-class package that weighs x ounces. In August of 2009, the cost was $1.22 for a package that weighed up to 1 ounce, plus 17 cents for each additional ounce or portion thereof (up to 13 ounces). (Source: United States Postal Service.) (a) Sketch a graph of f(x). (b) How is this function similar to the greatest integer function? How is it different?

Set your calculator to DOT mode and graph the greatest integer function, y = int(x), in the window [-4.7, 4.7] by [-3.1, 3.1] . Then complete the following analysis.

True or False The greatest integer function has an inverse function. Justify your answer.

True or False The logistic function has two horizontal asymptotes. Justify your answer.

Which function has range {all real numbers}? (A) \(f(x)=4+\ln x\) (B) \(f(x)=3-1 / x\) (C) \(f(x)=5 /\left(1+e^{-x}\right)\) (D) \(f(x)=\operatorname{int}(x-2)\) (E) \(f(x)=4 \cos x\)

Which function is bounded both above and below? (A) \(f(x)=x^{2}-4\) (B) \(f(x)=(x-3)^{3}\) (C) \(f(x)=3 e^{x}\) (D) \(f(x)=3+1 /\left(1+e^{-x}\right)\) (E) \(f(x)=4-|x|\)

Which of the following is the same as the restricted-domain function \(f(x)=\text { int }(x), 0 \leq x<2 \text { ? }\) (A) \(f(x)=\left\{\begin{array}{ll}0 & \text { if } 0 \leq x<1 \\1 & \text { if } x=1 \\2 & \text { if } 1<x<2\end{array}\right.\) (B) \(f(x)=\left\{\begin{array}{ll}0 & \text { if } x=0 \\1 & \text { if } 0<x \leq 1 \\2 & \text { if } 1<x<2\end{array}\right.\) (C) \(f(x)=\left\{\begin{array}{ll}0 & \text { if } 0 \leq x<1 \\1 & \text { if } 1 \leq x<2\end{array}\right.\) (D) \(f(x)=\left\{\begin{array}{ll}1 & \text { if } 0 \leq x<1 \\2 & \text { if } 1 \leq x<2\end{array}\right.\) (E) \(f(x)=\left\{\begin{array}{ll}x & \text { if } 0 \leq x<1 \\1+x & \text { if } 1 \leq x<2\end{array}\right.\)

Which function is increasing on the interval \((-\infty, \infty)\)? (A) \(f(x)=\sqrt{3+x}\) (B) \(f(x)=\operatorname{int}(x)\) (C) \(f(x)=2 x^{2}\) (D) \(f(x)=\sin x\) (E) \(f(x)=3 /\left(1+e^{-x}\right)\)

For positive values of x, we wish to compare the values of the basic functions x2, x, and \(\sqrt(x)\) (a) How would you order them from least to greatest? (b) Graph the three functions in the viewing window [0, 30] by [0, 20]. Does the graph confirm your response in (a)? (c) Now graph the three functions in the viewing window [0, 2] by [0, 1.5]. (d) Write a careful response to the question in (a) that accounts for all positive values of x.

Odds and Evens There are four odd functions and three even functions in the gallery of twelve basic functions. After multiplying these functions together pairwise in different combinations and exploring the graphs of the products, make a conjecture about the symmetry of: (a) a product of two odd functions; (b) a product of two even functions; (c) a product of an odd function and an even function.

Group Activity Assign to each student in the class the name of one of the twelve basic functions, but secretly so that no student knows the name of another. (The same function name could be given to several students, but all the functions should be used at least once.) Let each student make a one sentence self-introduction to the class that reveals something personal about who I am that really identifies me. The rest of the students then write down their guess as to the functions identity. Hints should be subtle and cleverly anthropomorphic. (For example, the absolute value function saying I have a very sharp smile is subtle and clever, while I am absolutely valuable is not very subtle at all.)

Pepperoni Pizzas For a statistics project, a student counted the number of pepperoni slices on pizzas of various sizes at a local pizzeria, compiling the following table: (a) Explain why the pepperoni count (P) ought to be proportional to the square of the radius (r). (b) Assuming that \(P=k \cdot r^{2}\), use the data pair (4, 12) to find the value of k. (c) Does the algebraic model fit the rest of the data well? (d) Some pizza places have charts showing their kitchen staff how much of each topping should be put on each size of pizza. Do you think this pizzeria uses such a chart? Explain.

Inverse Functions Two functions are said to be inverses of each other if the graph of one can be obtained from the graph of the other by reflecting it across the line . For example, the functions with the graphs shown below are inverses of each other: (a) Two of the twelve basic functions in this section are inverses of each other. Which are they? (b) Two of the twelve basic functions in this section are their own inverses. Which are they? (c) If you restrict the domain of one of the twelve basic functions to [0,\(infty\)) it becomes the inverse of another one. Which are they?

Identifying a Function by Its Properties (a) Seven of the twelve basic functions have the property that f(0) = 0. Which five do not? (b) Only one of the twelve basic functions has the property that \(f(x+y)=f(x)+f(y)\) for all x and y in its domain. Which one is it? (c) One of the twelve basic functions has the property that f(x+y)=f(x) f(y) for all x and y in its domain. Which one is it? (d) One of the twelve basic functions has the property that f(x y)=f(x)+f(y) for all x and y in its domain. Which one is it? (e) Four of the twelve basic functions have the property that \(f(x)+f(-x)=0\) for all x in their domains. Which four are they?