Chapter 1

(a) \(f(x)=\frac{1}{x+2}\) (b) \(g(x)=\frac{5}{x^{2}+4}\)

Which of the following rules can be modeled as a function? If a rule is not a function, explain why not. (a) For a particular flask, the rule assigns to every volume (input) of liquid in the flask the corresponding height (output). (b) For a particular flask, the rule assigns to every height (input) the corresponding volume (output). (c) The rule assigns to every person his or her birthday. (d) The rule assigns to every recorded singer the title of his or her first recorded song. (e) The rule assigns to every state the current representative in the House of Representatives. (f) The rule assigns to every current member of the House of Representatives the state he or she represents. (g) The rule assigns to every number the square of that number. (h) The rule assigns to every nonzero number the reciprocal of that number.

(a) \(f(x)=\sqrt{x}\) (b) \(g(x)=\sqrt{x-3}\) (c) \(h(x)=\sqrt{x^{2}-4}\)

(a) \(f(x)=\sqrt{\frac{1}{x+1}}\) (b) \(g(x)=\sqrt{\frac{x}{x+1}}\)

There are infinitely many prime numbers. This has been known for a long time; Euclid proved it sometime between 300 B.C. and 200 B.C. \({ }^{6}\) Number theorists (mathematicians who study the theory and properties of numbers) are interested in the distribution of prime numbers. Let \(P(n)=\) number of primes less than or equal to \(n\), where \(n\) is a positive number. Is \(P(n)\) a function? Explain.

Writing: We would like to tailor this course to your needs and interests; therefore we'd like to find out more about what these needs and interests are. On a sheet of paper separate from the rest of your homework, please write a paragraph or two telling us a bit about yourself by addressing the following questions. (a) What are you interested in studying in the future, both in terms of math and otherwise? (b) Are there things you have found difficult or confusing in mathematics in the past? If so, what? (c) What was your approach to studying mathematics in the past? Did it work well for you? (d) What are your major extracurricular activities or interests? (e) What do you hope to get out of this course?

Two functions are equivalent if they have the same domain and the same input/output relationship. The first function listed on each line below is called \(f .\) Which of the functions listed on each line are equivalent to \(f ?\) The domain of each function is the set of all real numbers. (Be careful to think about the sign of each function.) \(\begin{array}{lll}\text { (a) } f(x)=-2 x^{2} & g(w)=(-2 w)^{2} & i(t)=2\left(-t^{2}\right)\end{array}\) \(j(x)=-\sqrt{2 x^{2}} \sqrt{x^{2}} \sqrt{2}\) (b) \(f(x)=(2 x-1)^{2} \quad g(c)=(1-2 c)^{2} \quad h(t)=1-2 t^{2} \quad j(x)=1-(2 x)^{2}\) (c) \(f(x)=\sqrt{x^{2}}\) \(\lambda(m)=m\) \(\mathcal{T}(x)=\sqrt{(-x)^{2}} \quad \varphi(s)=|s|\)

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline 0 & 2 \\ 1 & 3 \\ 2 & 2 \\ 3 & 3 \\ 4 & 2 \end{array} $$

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline \sqrt{2} & 2 \\ \sqrt{3} & 3 \\ \sqrt{5} & 5 \\ \sqrt{6} & 6 \end{array} $$

For Problems 7 through 9 determine whether the relationship described is a function. If the relationship is a function, (a) what is the domain? the range? (b) is the function 1 -to- 1 ? $$ \begin{array}{ll} \text { Input } & \text { Output } \\ \hline \sqrt{2} & 0 \\ 2 \sqrt{2} & 0 \\ 3 \sqrt{2} & 0 \\ 4 \sqrt{2} & 0 \end{array} $$

Express each of the following rules for obtaining the output of a function using functional notation. (a) Square the input, add 3 , and take the square root of the result. (b) Double the input, then add 7 . (c) Take half of 3 less than the input. (d) Increase the input by 10, then cube the result.

Let \(C\) be a circle of radius 1 and let \(A(n)\) be the area of a regular \(n\) -gon inscribed in the circle. For instance, \(A(3)\) is the area of an equilateral triangle inscribed in circle \(C, A(4)\) is the area of a square inscribed in circle \(C\), and \(A(5)\) is the area of a regular pentagon inscribed in circle \(C\). (A polygon inscribed in a circle has all its vertices lying on the circle. A regular polygon is a polygon whose sides are all of equal length and whose angles are all of equal measure.) (a) Find \(A(4)\). (b) Is \(A(n)\) a function? If it is, answer the questions that follow. (c) What is the natural domain of \(A(n)\) ? (d) As \(n\) increases, do you think that \(A(n)\) increases, or decreases? This is hard to justify rigorously, but what does your intuition tell you? (e) Will \(A(n)\) increase without bound as \(n\) increases, or is there a lid above which the values of \(A(n)\) will never go? If there is such a lid (called an upper bound) give one. What is the smallest lid possible? Rigorous justification is not requested.

Some friends are taking a long car trip. They are traveling east on Route 66 from Flagstaff, Arizona, through New Mexico and Texas and into Oklahoma. Let \(f\) be the function that gives the number of miles traveled \(t\) hours into the trip, where \(t=0\) denotes the beginning of the trip. For instance, \(f(7)\) is the mileage 7 hours into the trip. If the travelers set an odometer to zero at the start of the trip, the output of \(f\) would be the reading on the odometer. Let \(g\) be the function that gives the car's speed \(t\) hours into the trip, where \(t=0\) denotes the beginning of the trip. For instance, \(g(7)\) is the car's speed 7 hours into the trip. The output of \(g\) corresponds to the speedometer reading. Suppose they pass a sign that reads "entering Gallup, New Mexico," \(h\) hours into the trip. (a) Write the following expressions using functional notation wherever appropriate. i. The car's speed 1 hour before reaching Gallup ii. 10 miles per hour slower than the speed of the car entering Gallup iii. Half the time it took to reach Gallup iv. Their speed 6 hours after reaching Gallup v. The distance traveled in the first 2 hours of the trip vi. The distance traveled in the second 2 hours of the trip vii. Half the distance covered in the second 3 hours of travel viii. The average speed in the first 5 hours of travel (Average speed is computed by dividing the distance traveled by the time elapsed.) ix. The average speed between hour 6 of the trip and hour 12 of the trip (b) Interpret the following in words. i. \(f(h+2)\) ii. \(\frac{1}{2} f(h)\) iii. \(f\left(\frac{h}{2}\right)\) iv. \(f(h-2)\) v. \(f(h)-2\) vi. \(f(h)+2\) vii. \(g(h+2)\) viii. \(g(h)+2\) ix. \(g(h)-2\) x. \(\frac{1}{2} g(h)\) xi. \(\frac{1}{2} g(h-1)\)

Let \(C(w)\) be the amount (in dollars) it costs you to mail your grandmother a firstclass package weighing \(w\) ounces. Suppose you just mailed her a birthday present that weighed \(A\) ounces. Describe in words the practical meaning of each of the following expressions. (a) \(C(A)\) (b) \(C(2 A)\) (c) \(2 C(A)\) (d) \(C(A+1)\) (e) \(C(A)+1\)

If \(f(x)=\sqrt{\frac{1}{x+1}}\), find the following. Simplify your answer where possible. (a) \(f(0)\) (b) \(f(3)\) (c) \(f\left(-\frac{1}{4}\right)\) (d) \(f(b)\) (e) \(f(b-1)\) (f) \(f(b+3)\) (g) \([f(7)]^{2}\) (h) \(f\left(b^{2}\right)\) (i) \([f(b)]^{2}\)

Draw the graph of a function \(f\) that is 1 -to- 1 and a function \(g\) that is not 1 -to-1.

If \(g(x)=\frac{\sqrt{x^{2}+4}}{2}\), find the following. Simplify your answer where possible. (a) \(g(0)\) (b) \(g(2)\) (c) \(g(\sqrt{5})\) (d) \(g\left(\frac{1}{\sqrt{2}}\right)\) (e) \(-g(3 t)\) (f) \(g(\sqrt{t-4})\)

If \(h(x)=\frac{x^{2}}{1-2 x}\), find (a) \(h(0)\) (b) \(h(3)\) (c) \(h(p+1)\) (d) \(h(3 p)\) (e) \(2 h(3 p)\) (f) \(\frac{1}{h(2 p)}\)

If \(j(x)=3 x^{2}-2 x+1\), find the following. Simplify your answer where possible. (a) \(j(0)\) (b) \(j(1)\) (c) \(j(-1)\) (d) \(j(-x)\) (e) \(j(x+2)\) (f) \(3 j(x)\) (g) \(j(3 x)\)

In Problems 19 through 21 (a) find the value of the function at \(x=0, x=1\), and \(x=-1\). (b) find all \(x\) such that the value of the function is \((i) 0,(i i) 1\), and \((i i i)-1 .\) $$ f(x)=\frac{3 x+5}{2} $$

(a) find the value of the function at \(x=0, x=1\), and \(x=-1\). (b) find all \(x\) such that the value of the function is \((i) 0,(i i) 1\), and \((i i i)-1 .\) $$ g(x)=x^{2}-1 $$

(a) find the value of the function at \(x=0, x=1\), and \(x=-1\). (b) find all \(x\) such that the value of the function is \((i) 0,(i i) 1\), and \((i i i)-1 .\) $$ h(x)=\frac{x^{2}+2 x}{2 x+2} $$ (For a review of quadratic equations, refer to the Algebra Appendix.)

For each function in Problems 22 through 27, determine the largest possible domain. (a) \(f(x)=\frac{1}{5 x+10}\) (b) \(g(x)=\sqrt{5 x+10}\)

For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x^{2}-1}\) (b) \(g(x)=\sqrt{x^{2}-1}\) For part (b), factor the quadratic. The product must be positive. For more assistance, refer to the Algebra Appendix.

For each function, determine the largest possible domain. (a) \(f(x)=\frac{3}{x^{2}+3 x-4}\) (b) \(g(x)=\sqrt{x^{2}+3 x-4}\) Factoring will help clarify the solution.

For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x^{2}+2 x+1}\) (b) \(g(x)=\sqrt{x^{2}+2 x+1}\) Factoring will help clarify the solution.

Express the area of a circle as a function of: (a) its diameter, \(d\). (b) its circumference, \(c .\)

For each function, determine the largest possible domain. (a) \(f(x)=\frac{1}{x+2}-\frac{1}{x-1}\) (b) \(g(x)=\sqrt{x+2}-\sqrt{x-1}\)

Express the surface area of a cube as a function of the length \(s\) of one side.

For each function, determine the largest possible domain. (a) \(f(x)=\frac{3}{x}+\frac{2}{3-x}-\frac{x}{2 x+2}\) (b) \(g(x)=\sqrt{x}+2 \sqrt{3-x}\)

A closed rectangular box has a square base. Let \(s\) denote the length of the sides of the base and let \(h\) denote the height of the box, \(s\) and \(h\) in inches. (a) Express the volume of the box in terms of \(s\) and \(h\). (b) Express the surface area of the box in terms of \(s\) and \(h\). (c) If the volume of the box is 120 cubic inches, express the surface area of the box as a function of \(s\).

The volume of a sphere and the surface area of a sphere are both functions of the sphere's radius. The volume function is given by \(V(r)=\frac{4}{3} \pi r^{3}\) and the surface area function is given by \(S(r)=4 \pi r^{2}\). (a) If the radius of a sphere is doubled, by what factor is the volume multiplied? The surface area? (b) Which results in a larger increase in surface area: increasing the radius of a sphere by 1 unit or increasing the surface area by 12 units? Does the answer depend upon the original radius of the sphere? Explain your reasoning completely. (It may be useful to check your answer in a specific case as a spot check for errors.) (c) In order to double the surface area of the sphere, by what factor must the radius be multiplied? (d) In order to double the volume of the sphere, by what factor must the radius be multiplied?

You are constructing a closed rectangular box with a square base and a volume of 200 cubic inches. If the material for the base and lid costs 10 cents per square inch and the material for the sides costs 7 cents per square inch, express the cost of material for the box as a function of \(s\), the length of the side of the base.

Let \(A=A(S)\) be the area function for a square of side \(S .\) A takes as input the length \(S\) of the side of a square and gives as output the area of the square. (a) Find the following. \(\begin{array}{llll}\text { i. } A(4) & \text { ii. } A(W) & \text { iii. } A(\sqrt{2}+3) & \text { iv. } A(4+h) & \text { v. } A(x-1)\end{array}\) (b) Suppose that \(S\) is bigger than 1 . Which is larger, \(A(S-1)\) or \(A(S)-1 ?\) (Does the answer depend on the size of \(S ?\) If so, how?) (c) Explain in words the difference between \(A(S-1)\) and \(A(S)-1\). Which one of the expressions corresponds to the shaded area in the accompanying figure? What area corresponds to the other expression?

A rectangle is inscribed in a semicircle of radius \(R\), where \(R\) is constant. (a) Express the area of the rectangle as a function of the height, \(h\), of the rectang \(A=f(h)\) (b) Express the perimeter of the rectangle as a function of the height, \(h\), of rectangle, \(P=g(h)\).

Two sisters, Nina and Lori, part on a street corner. Lori saunters due north at a rate of 150 feet per minute and Nina jogs off due east at a rate of 320 feet per minute. Assuming they maintain their speeds and directions, express the distance between the sisters as a function of the number of minutes since they parted.

For Problems 33 through 35, if the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all cases, indicate the interval on the number line. $$ \text { (a) }-1 \leq x \leq 3 $$

If the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all cases, indicate the interval on the number line. $$ \text { (a) }-7 \leq x<-5 $$

If the interval is written using inequalities, write it using interval notation; if it is expressed in interval notation, rewrite it using inequalities. In all cases, indicate the interval on the number line. $$ \text { (a) }-1

Let \(f(x)=2 x^{2}+x\). Find the following. (a) \(f(3)\) (b) \(f(2 x)\) (c) \(f(1+x)\) (d) \(f\left(\frac{1}{x}\right)\) (e) \(\frac{1}{f(x)}\)

A right circular cylinder is inscribed in a sphere of radius \(5 .\) (a) Express the volume of the cylinder as a function of its radius, \(r\). (b) Express the surface area of the cylinder as a function of its radius, \(r\)

The height of a right circular cone is one third of the diameter of the base. (a) Express its volume as a function of its height, \(h\). (b) Express its volume as a function of \(r\), the radius of its base.

A vitamin capsule is constructed from a cylinder of length \(x\) centimeters and radius \(r\) centimeters, capped on either end by a hemisphere, as shown at left. Suppose that the length of the cylinder is equal to three times the diameter of the hemispherical caps. (a) Express the volume of the vitamin capsule as a function of \(x .\) Your strategy should be to begin by expressing the volume as a function of both \(x\) and \(r\). (b) Express the surface area of the vitamin capsule as a function of \(x\).

Assume that \(f\) is a function with domain \((-\infty, \infty)\). Which of the following statements is true for every such function \(f\) and all \(p, w\), and \(z\) in the domain of \(f ?\) If a statement is not true for every function, find a function for which it is false. (Hint: The constant functions are good functions to use as a first check.) (a) \(f(2) f(3)=f(6)\) (b) \(f(p)+f(p)=2 f(p)\) (c) \(f(4+5)=f(4)+f(5)\) (d) \(f(w) f(w)=[f(w)]^{2}\) (e) \(f(z) f(z)=f\left(z^{2}\right)\) (f) \(\frac{f(x)}{5[f(x)]^{2}}=\frac{1}{5 f(x)}\), where \(f(x) \neq 0\) (g) \(f(1) f(7)=f(7)\)

Translate each of these English sentences into a mathematical sentence, i.e., an equation. Then comment on the validity of the statement, making qualifications if necessary. You will need to define your variables. The first example has been worked for you. Example: The cost of broccoli is proportional to its weight. Answer: Let \(C=\) the cost of broccoli and \(B=\) the weight of broccoli. \(C=k B \quad\) for some constant \(k\). We can emphasize that cost is a function of weight by writing $$ C(B)=k B . $$ Validity: In general this is true, because broccoli is usually sold by the pound. The proportionality constant \(k\) is the price per pound; at any given time and store it is fixed. The proportionality constant may vary with the season and the specific store. (a) The cost of a piece of sculpture is proportional to its weight. (b) The rate at which money grows in a savings account is proportional to the amount of money in the account. (c) The rate at which a population grows is proportional to the size of the population. (d) The total distance you travel is proportional to the time you spend traveling.

A gardener has a fixed length of fence that she will use to fence off a rectangular chili pepper garden. Express the area of the garden as a function of the length of one side of the garden. If you have trouble, reread the "Portable Strategies for Problem Solving" listed in this chapter. We've also included the following advice geared specifically toward this particular problem. Give the length of fencing a name, such as \(L\). (We don't know what \(L\) is, but we know that it is fixed, so \(L\) is a constant, not a variable.) \- Draw a picture of the garden. Call the length of one side of the fence \(s\). How can you express the length of the adjacent side in terms of \(L\) and \(s ?\) \- What expression gives the area enclosed by the fence?

Cathy will fence off a circular pen for her rabbits. Express the area of the rabbit pen as a function of the length of fencing she uses.

A commuter rides his bicycle to the train station, takes the subway downtown and then walks from the subway station to his office. He bikes at an average speed of \(B\) miles per hour and can walk \(M\) miles in \(H\) hours. The subway ride takes \(R\) minutes. The commuter bikes \(X\) miles and walks \(Y\) miles to get to work. Assume that \(X, Y, B, H\), \(M\), and \(R\) are all constants. The amount of time it takes the commuter to get to work varies with how long he has to spend at the subway station locking his bike and waiting for the next train. Denote this time by \(w\), where \(w\) is in hours. Express the time it takes for his commute as a function of \(w\). Specify whether your answer is in minutes or in hours.

During road construction gravel is being poured onto the ground from the top of a tall truck. The gravel falls into a conical pile whose height is always equal to half of its radius. Express the amount of gravel in the pile as a function of its (a) height. (b) radius.

(a) A bead maker has a collection of wooden spheres 2 centimeters in diameter and is making beads by drilling holes through the center of each sphere. The length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. If you are stuck, begin by trying to express half the length of the bead as a function of the radius of the hole drilled. (b) More generally, suppose he works with spherical beads of radius \(\mathrm{R}\). Again, the length of the bead is a function of the diameter of the hole he drills. Find a formula for this function. In the following problems, demonstrate your use of the portable strategies for problem solving described in this chapter. What simpler questions are you asking yourself? What concrete example can you give to convince your friends and relatives that you are right? Write this up clearly, so a reader can follow your train of thought easily.