Chapter 1: Functions

1.1 Functions and Their Graphs

Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table, or a verbal description; we will use all four representations throughout this text. This section reviews these ideas.

Functions; Domain and Range

The temperature at which water boils depends on the elevation above sea level. The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The distance an object travels depends on the elapsed time.

In each case, the value of one variable quantity, say y, depends on the value of another variable quantity, which we often call x. We say that “y is a function of x” and write this symbolically as

The symbol f represents the function, the letter x is the independent variable representing the input value to f, and y is the dependent variable or output value of f at x.

DEFINITION A function f from a set D to a set Y is a rule that assigns a single value 𝑓(𝑥) in Y to each x in D.

A rule that assigns more than one value to an input x, such as the rule that assigns to a positive number both the positive and negative square roots of the number, does not describe a function.

The set D of all possible input values is called the domain of the function. The domain of f will sometimes be denoted by 𝐷(𝑓) . The set of all output values 𝑓(𝑥) as x varies throughout D is called the range of the function. The range might not include every element in the set Y. The domain and range of a function can be any sets of objects, but often in calculus they are sets of real numbers interpreted as points of a coordinate line. (In Chapters 12–15, we will encounter functions for which the elements of the sets are points in the plane, or in space.)

Often a function is given by a formula that describes how to calculate the output value from the input variable. For instance, the equation 𝐴 =𝜋𝑟2 is a rule that calculates the area A of a circle from its radius r. When we define a function f with a formula 𝑦 =𝑓(𝑥) and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of real x-values for which the formula gives real y-values. This is called the natural domain of f. If we want to restrict the domain in some way, we must say so. The domain of 𝑦 =𝑥2 is the entire set of real numbers. To restrict the domain of the function to, say, positive values of x, we would write “ 𝑦 =𝑥2,𝑥 >0 .”

FIGURE 1.1 A diagram showing a function as a kind of machine.

FIGURE 1.2 A function from a set D to a set Y assigns a unique element of Y to each element in D.

Changing the domain to which we apply a formula usually changes the range as well. The range of 𝑦 =𝑥2 is [0,∞) . The range of 𝑦 =𝑥2,𝑥 ≥2 , is the set of all numbers obtained by squaring numbers greater than or equal to 2. In set notation (see Appendix A.1), the range is {𝑥2|𝑥 ≥2} or {𝑦|𝑦 ≥4} or [4,∞) .

When the range of a function is a set of real numbers, the function is said to be real-valued. The domains and ranges of most real-valued functions we consider are intervals or combinations of intervals. Sometimes the range of a function is not easy to find.

A function f is like a machine that produces an output value 𝑓(𝑥) in its range whenever we feed it an input value x from its domain (Figure 1.1). The function keys on a calculator give an example of a function as a machine. For instance, whenever you enter a nonnegative number x and press the √𝑥 key, the calculator gives an output value (the square root of x).

A function can also be pictured as an arrow diagram (Figure 1.2). Each arrow associates to an element of the domain D a single element in the set Y. In Figure 1.2, the arrows indicate that 𝑓(𝑎) is associated with a, 𝑓(𝑥) is associated with x, and so on. Notice that a function can have the same output value for two different input elements in the domain (as occurs with 𝑓(𝑎) in Figure 1.2), but each input element x is assigned a single output value 𝑓(𝑥) .

EXAMPLE 1 Verify the natural domains and associated ranges of some simple functions. The domains in each case are the values of x for which the formula makes sense.

Solution The formula 𝑦 =𝑥2 gives a real y-value for any real number x, so the domain is ( −∞,∞) . The range of 𝑦 =𝑥2 is [0,∞) because the square of any real number is non-negative and every nonnegative number y is the square of its own square root: 𝑦 =(√𝑦)2 .

The formula 𝑦 =1/𝑥 gives a real 𝑦 -value for every 𝑥 except 𝑥 =0 . For consistency in the rules of arithmetic, we cannot divide any number by zero. The range of 𝑦 =1/𝑥 , the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since 𝑦 =1/(1/𝑦) . That is, for 𝑦 ≠0 the number 𝑥 =1/𝑦 is the input that is assigned to the output value 𝑦 .

The formula 𝑦 =√𝑥 gives a real 𝑦 -value only if 𝑥 ≥0 . The range of 𝑦 =√𝑥 is [0,∞) because every nonnegative number is some number’s square root (namely, it is the square root of its own square).

In 𝑦 =√4−𝑥 , the quantity 4 −𝑥 cannot be negative. That is, 4 −𝑥 ≥0 , or 𝑥 ≤4 . The formula gives nonnegative real 𝑦 -values for all 𝑥 ≤4 . The range of √4−𝑥 is [0,∞) , the set of all nonnegative numbers.

The formula 𝑦 =√1−𝑥2 gives a real y-value for every x in the closed interval from -1 to 1. Outside this domain, 1 −𝑥2 is negative and its square root is not a real number. The values of 1 −𝑥2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of √1−𝑥2 is [0,1].

Graphs of Functions

If f is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for f. In set notation, the graph is

The graph of the function 𝑓(𝑥) =𝑥 +2 is the set of points with coordinates (𝑥,𝑦) for which 𝑦 =𝑥 +2 . Its graph is the straight line sketched in Figure 1.3.

The graph of a function f is a useful picture of its behavior. If (𝑥,𝑦) is a point on the graph, then 𝑦 =𝑓(𝑥) is the height of the graph above (or below) the point x. The height may be positive or negative, depending on the sign of 𝑓(𝑥) (Figure 1.4).

FIGURE 1.3 The graph of 𝑓(𝑥) =𝑥 +2 is the set of points (𝑥,𝑦) for which y has the value 𝑥 +2 .

FIGURE 1.4 If (𝑥,𝑦) lies on the graph of f, then the value 𝑦 =𝑓(𝑥) is the height of the graph above the point x (or below x if 𝑓(𝑥) is negative).

EXAMPLE 2 Graph the function 𝑦 =𝑥2 over the interval [ −2,2] .

Solution Make a table of xy-pairs that satisfy the equation 𝑦 =𝑥2 . Plot the points (𝑥,𝑦) whose coordinates appear in the table, and draw a smooth curve (labeled with its equation) through the plotted points (see Figure 1.5).

FIGURE 1.5 Graph of the function in Example 2.

How do we know that the graph of 𝑦 =𝑥2 doesn’t look like one of these curves?

To find out, we could plot more points. But how would we then connect them? The basic question still remains: How do we know for sure what the graph looks like between the points we plot? Calculus answers this question, as we will see in Chapter 4. Meanwhile, we will have to settle for plotting points and connecting them as best we can.

Representing a Function Numerically

A function may be represented algebraically by a formula and visually by a graph (Example 2). Another way to represent a function is numerically, through a table of values. From an appropriate table of values, a graph of the function can be obtained using the method illustrated in Example 2, possibly with the aid of a computer. The graph consisting of only the points in the table is called a scatterplot.

EXAMPLE 3 Musical notes are pressure waves in the air. The data associated with Figure 1.6 give recorded pressure displacement versus time in seconds of a musical note produced by a tuning fork. The table provides a representation of the pressure function (in micropascals) over time. If we first make a scatterplot and then draw a smooth curve that approximates the data points (𝑡,𝑝) from the table, we obtain the graph shown in the figure.

FIGURE 1.6 A smooth curve approximating the plotted points gives a graph of the pressure function represented by the accompanying tabled data (Example 3).

The Vertical Line Test for a Function

Not every curve in the coordinate plane can be the graph of a function. A function f can have only one value 𝑓(𝑥) for each x in its domain, so no vertical line can intersect the graph of a function at more than one point. If a is in the domain of the function f, then the vertical line x = a will intersect the graph of f at the single point (𝑎,𝑓(𝑎)) .

A circle cannot be the graph of a function, since some vertical lines intersect the circle twice. The circle graphed in Figure 1.7a, however, contains the graphs of two functions of x, namely the upper semicircle defined by the function 𝑓(𝑥) =√1−𝑥2 and the lower semicircle defined by the function 𝑔(𝑥) = −√1−𝑥2 (Figures 1.7b and 1.7c).

(a) 𝑥2 +𝑦2 =1

(b) 𝑦 =√1−𝑥2

(c) 𝑦 = −√1−𝑥2

FIGURE 1.7 (a) The circle is not the graph of a function; it fails the vertical line test. (b) The upper semicircle is the graph of the function 𝑓(𝑥) =√1−𝑥2 . (c) The lower semicircle is the graph of the function 𝑔(𝑥) = −√1−𝑥2 .

FIGURE 1.8 The absolute value function has domain ( −∞,∞) and range [0,∞) .

FIGURE 1.9 To graph the function 𝑦 =𝑓(𝑥) shown here, we apply different formulas to different parts of its domain (Example 4).

FIGURE 1.10 The graph of the greatest integer function 𝑦 =⌊𝑥⌋ lies on or below the line y = x, so it provides an integer floor for x (Example 5).

FIGURE 1.11 The graph of the least integer function 𝑦 =⌈𝑥⌉ lies on or above the line y = x, so it provides an integer ceiling for x (Example 6).

Piecewise-Defined Functions

Sometimes a function is described in pieces by using different formulas on different parts of its domain. One example is the absolute value function

whose graph is given in Figure 1.8. The right-hand side of the equation means that the function equals x if 𝑥 ≥0 , and equals -x if x < 0. Piecewise-defined functions often arise when real-world data are modeled. Here are some other examples.

EXAMPLE 4 The function

is defined on the entire real line but has values given by different formulas, depending on the position of x. The values of f are given by y = -x when x < 0, 𝑦 =𝑥2 when 0 ≤𝑥 ≤1 , and y = 1 when x > 1. The function, however, is just one function whose domain is the entire set of real numbers (Figure 1.9).

EXAMPLE 5 The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted ⌊𝑥⌋ . Figure 1.10 shows the graph. Observe that

EXAMPLE 6 The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted [x]. Figure 1.11 shows the graph. For positive values of x, this function might represent, for example, the cost of parking x hours in a parking lot that charges $1 for each hour or part of an hour.

Increasing and Decreasing Functions

If the graph of a function climbs or rises as you move from left to right, we say that the function is increasing. If the graph descends or falls as you move from left to right, the function is decreasing.

DEFINITIONS Let f be a function defined on an interval I and let 𝑥1 and 𝑥2 be two distinct points in I.

1. If 𝑓(𝑥2) >𝑓(𝑥1) whenever 𝑥1 <𝑥2 , then f is said to be increasing on I.

2. If 𝑓(𝑥2) <𝑓(𝑥1) whenever 𝑥1 <𝑥2 , then f is said to be decreasing on I.

It is important to realize that the definitions of increasing and decreasing functions must be satisfied for every pair of points 𝑥1 and 𝑥2 in I with 𝑥1 <𝑥2 . Because we use the inequality < to compare the function values, instead of ≤ , it is sometimes said that f is strictly increasing or decreasing on I. The interval I may be finite (also called bounded) or infinite (unbounded).

(a)

FIGURE 1.12 (a) The graph of 𝑦 =𝑥2 (an even function) is symmetric about the y-axis. (b) The graph of 𝑦 =𝑥3 (an odd function) is symmetric about the origin.

EXAMPLE 7 The function graphed in Figure 1.9 is decreasing on ( −∞,0) and increasing on (0,1) . The function is neither increasing nor decreasing on the interval (1,∞) because the function is constant on that interval, and hence the strict inequalities in the definition of increasing or decreasing are not satisfied on (1,∞) .

Even Functions and Odd Functions: Symmetry

The graphs of even and odd functions have special symmetry properties.

DEFINITIONS A function 𝑦 =𝑓(𝑥) is an even function of 𝑥 if 𝑓( −𝑥) =𝑓(𝑥) , odd function of 𝑥 if 𝑓( −𝑥) = −𝑓(𝑥) , for every 𝑥 in the function’s domain.

The names even and odd come from powers of x. If y is an even power of x, as in 𝑦 =𝑥2 or 𝑦 =𝑥4 , it is an even function of x because ( −𝑥)2 =𝑥2 and ( −𝑥)4 =𝑥4 . If y is an odd power of x, as in y = x or 𝑦 =𝑥3 , it is an odd function of x because ( −𝑥)1 = −𝑥 and ( −𝑥)3 = −𝑥3 .

The graph of an even function is symmetric about the y-axis. Since 𝑓( −𝑥) =𝑓(𝑥) , a point (𝑥,𝑦) lies on the graph if and only if the point ( −𝑥,𝑦) lies on the graph (Figure 1.12a). A reflection across the y-axis leaves the graph unchanged.

The graph of an odd function is symmetric about the origin. Since 𝑓( −𝑥) = −𝑓(𝑥) , a point (𝑥,𝑦) lies on the graph if and only if the point ( −𝑥, −𝑦) lies on the graph (Figure 1.12b). Equivalently, a graph is symmetric about the origin if a rotation of 180∘ about the origin leaves the graph unchanged.

Notice that each of these definitions requires that both x and -x be in the domain of f.

EXAMPLE 8 Here are several functions illustrating the definitions.

Even function: ( −𝑥)2 =𝑥2 for all x; symmetry about y-axis. So 𝑓( −3) =9 =𝑓(3) . Changing the sign of x does not change the value of an even function.

Even function: ( −𝑥)2 +1 =𝑥2 +1 for all x; symmetry about y-axis (Figure 1.13a).

(a)

(b)

FIGURE 1.13 (a) When we add the constant term 1 to the function 𝑦 =𝑥2 , the resulting function 𝑦 =𝑥2 +1 is still even and its graph is still symmetric about the y-axis. (b) When we add the constant term 1 to the function y = x, the resulting function 𝑦 =𝑥 +1 is no longer odd, since the symmetry about the origin is lost. The function 𝑦 =𝑥 +1 is also not even (Example 8).

𝑓(𝑥) =𝑥 Odd function: ( −𝑥) = −𝑥 for all 𝑥 ; symmetry about the origin. So 𝑓( −3) = −3 while 𝑓(3) =3 . Changing the sign of 𝑥 changes the sign of the value of an odd function.

𝑓(𝑥) =𝑥 +1 Not odd: 𝑓( −𝑥) = −𝑥 +1 , but −𝑓(𝑥) = −𝑥 −1 . The two are not equal.

Common Functions

A variety of important types of functions are frequently encountered in calculus.

Linear Functions A function of the form 𝑓(𝑥) =𝑚𝑥 +𝑏 , where m and b are fixed constants, is called a linear function. Figure 1.14a shows an array of lines 𝑓(𝑥) =𝑚𝑥 . Each of these has b = 0, so these lines pass through the origin. The function 𝑓(𝑥) =𝑥 , where m = 1 and b = 0, is called the identity function. Constant functions result when the slope is m = 0 (Figure 1.14b).

(b)

FIGURE 1.14 (a) Lines through the origin with slope m. (b) A constant function with slope m = 0.

DEFINITION Two variables y and x are proportional (to one another) if one is always a constant multiple of the other—that is, if y = kx for some nonzero constant k.

If the variable y is proportional to the reciprocal 1/x, then sometimes it is said that y is inversely proportional to x (because 1/x is the multiplicative inverse of x).

Power Functions A function 𝑓(𝑥) =𝑥𝑎 , where a is a constant, is called a power function. There are several important cases to consider.

(a) 𝑓(𝑥) =𝑥𝑎 with a = n, a positive integer.

The graphs of 𝑓(𝑥) =𝑥𝑛 , for n = 1, 2, 3, 4, 5, are displayed in Figure 1.15. These functions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x-axis on the interval ( −1,1) and to rise more steeply for |𝑥| >1 . Each curve passes through the point (1,1) and through the origin. The graphs of functions with even powers are symmetric about the y-axis; those with odd powers are symmetric about the origin. The even-powered functions are decreasing on the interval ( −∞,0] and increasing on [0,∞) ; the odd-powered functions are increasing over the entire real line ( −∞,∞) .

FIGURE 1.15 Graphs of 𝑓(𝑥) =𝑥𝑛,𝑛 =1,2,3,4,5 , defined for −∞ <𝑥 <∞ .

The graphs of the functions 𝑓(𝑥) =𝑥−1 =1/𝑥 and 𝑓(𝑥) =𝑥−2 =1/𝑥2 are shown in Figure 1.16. Both functions are defined for all 𝑥 ≠0 (you can never divide by zero). The graph of y = 1/x is the hyperbola xy = 1, which approaches the coordinate axes far from the origin. The graph of 𝑦 =1/𝑥2 also approaches the coordinate axes. The graph of the function 𝑓(𝑥) =1/𝑥 is symmetric about the origin; this function is decreasing on the intervals ( −∞,0) and (0,∞) . The graph of the function 𝑓(𝑥) =1/𝑥2 is symmetric about the y-axis; this function is increasing on ( −∞,0) and decreasing on (0,∞) .

FIGURE 1.16 Graphs of the power functions 𝑓(𝑥) =𝑥𝑎 . (a) 𝑎 = −1 . (b) 𝑎 = −2 .

The functions 𝑓(𝑥) =𝑥1/2 =√𝑥 and 𝑓(𝑥) =𝑥1/3 =3√𝑥 are the square root and cube root functions, respectively. The domain of the square root function is [0,∞) , but the cube root function is defined for all real x. Their graphs are displayed in Figure 1.17, along with the graphs of 𝑦 =𝑥3/2 and 𝑦 =𝑥2/3 . (Recall that 𝑥3/2 =(𝑥1/2)3 and 𝑥2/3 =(𝑥1/3)2 .)

FIGURE 1.17 Graphs of the power functions 𝑓(𝑥) =𝑥𝑎 for 𝑎 =12 , 13 , 32 , and 23 .

(b)

Polynomials A function p is a polynomial if

where n is a nonnegative integer and the numbers 𝑎0,𝑎1,𝑎2,…,𝑎𝑛 are real constants (called the coefficients of the polynomial). All polynomials have domain ( −∞,∞) . If the leading coefficient 𝑎𝑛 ≠0 , then n is called the degree of the polynomial. Linear functions with 𝑚 ≠0 are polynomials of degree 1. Polynomials of degree 2, usually written as 𝑝(𝑥) =𝑎𝑥2 +𝑏𝑥 +𝑐 , are called quadratic functions. Likewise, cubic functions are polynomials 𝑝(𝑥) =𝑎𝑥3 +𝑏𝑥2 +𝑐𝑥 +𝑑 of degree 3. Figure 1.18 shows the graphs of three polynomials. Techniques to graph polynomials are studied in Chapter 4.

FIGURE 1.18 Graphs of three polynomial functions.

Rational Functions A rational function is a quotient or ratio 𝑓(𝑥) =𝑝(𝑥)/𝑞(𝑥) , where p and q are polynomials. The domain of a rational function is the set of all real x for which 𝑞(𝑥) ≠0 . The graphs of three rational functions are shown in Figure 1.19.

FIGURE 1.19 Graphs of three rational functions. The straight red lines approached by the graphs are called asymptotes and are not part of the graphs. We discuss asymptotes in Section 2.5.

Algebraic Functions Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions (such as those satisfying an equation like 𝑦3 −9𝑥𝑦 +𝑥3 =0 , studied in Section 3.7). Figure 1.20 displays the graphs of three algebraic functions.

(b)

FIGURE 1.20 Graphs of three algebraic functions.

Trigonometric Functions The six basic trigonometric functions are reviewed in Section 1.3. The graphs of the sine and cosine functions are shown in Figure 1.21.

FIGURE 1.21 Graphs of the sine and cosine functions.

Exponential Functions A function of the form 𝑓(𝑥) =𝑎𝑥 , where a > 0 and 𝑎 ≠1 , is called an exponential function (with base a). All exponential functions have domain ( −∞,∞) and range (0,∞) , so an exponential function never assumes the value 0. We discuss exponential functions in Section 1.4. The graphs of some exponential functions are shown in Figure 1.22.

FIGURE 1.22 Graphs of exponential functions.

Logarithmic Functions These are the functions 𝑓(𝑥) =log𝑎⁡𝑥 , where the base 𝑎 ≠1 is a positive constant. They are the inverse functions of the exponential functions, and we discuss these functions in Section 1.5. Figure 1.23 shows the graphs of four logarithmic functions with various bases. In each case the domain is (0,∞) and the range is ( −∞,∞) .

FIGURE 1.23 Graphs of four logarithmic functions.

FIGURE 1.24 Graph of a catenary or hanging cable. (The Latin word catena means “chain.”)

Transcendental Functions These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions, and many other functions as well. The catenary is one example of a transcendental function. Its graph has the shape of a cable, like a telephone line or electric cable, strung from one support to another and hanging freely under its own weight (Figure 1.24). The function defining the graph is discussed in Section 7.3.

Exercises 1.1

Functions

In Exercises 1–6, find the domain and range of each function.

1. 𝑓(𝑥) =1 +𝑥2

2. 𝑓(𝑥) =1 −√𝑥

3. 𝐹(𝑥) =√5𝑥+10

4. 𝑔(𝑥) =√𝑥2−3𝑥

5. 𝑓(𝑡) =43−𝑡

6. 𝐺(𝑡) =2𝑡2−16

In Exercises 7 and 8, which of the graphs are graphs of functions of 𝑥 , and which are not? Give reasons for your answers.

7. a.

b.

8. a.

Finding Formulas for Functions

9. Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length 𝑥 .

10. Express the side length of a square as a function of the length 𝑑 of the square’s diagonal. Then express the area as a function of the diagonal length.

11. Express the edge length of a cube as a function of the cube’s diagonal length 𝑑 . Then express the surface area and volume of the cube as a function of the diagonal length.

12. A point P in the first quadrant lies on the graph of the function 𝑓(𝑥) =√𝑥 . Express the coordinates of P as functions of the slope of the line joining P to the origin.

13. Consider the point (𝑥,𝑦) lying on the graph of the line 2𝑥 +4𝑦 =5 . Let L be the distance from the point (𝑥,𝑦) to the origin (0,0) . Write L as a function of x.

14. Consider the point (𝑥,𝑦) lying on the graph of 𝑦 =√𝑥−3 . Let L be the distance between the points (𝑥,𝑦) and (4,0) . Write L as a function of y.

Functions and Graphs

Find the natural domain and graph the functions in Exercises 15-20.

15. 𝑓(𝑥) =5 −2𝑥

16. 𝑓(𝑥) =1 −2𝑥 −𝑥2

17. 𝑔(𝑥) =√|𝑥|

18. 𝑔(𝑥) =√−𝑥

19. 𝐹(𝑡) =𝑡/|𝑡|

20. 𝐺(𝑡) =1/|𝑡|

21. Find the domain of 𝑦 =𝑥+34−√𝑥2−9 .

22. Find the range of 𝑦 =2 +√9+𝑥2 .

23. Graph the following equations and explain why they are not graphs of functions of 𝑥 .

a. |𝑦| =𝑥

b. 𝑦2 =𝑥2

24. Graph the following equations and explain why they are not graphs of functions of 𝑥 .

a. |𝑥| +|𝑦| =1

b. |𝑥 +𝑦| =1

Piecewise-Defined Functions

Graph the functions in Exercises 25–28.

25. 𝑓(𝑥) ={𝑥,0≤𝑥≤12−𝑥,1<𝑥≤2

26. 𝑔(𝑥) ={1−𝑥,0≤𝑥≤12−𝑥,1<𝑥≤2

27. 𝐹(𝑥) ={4−𝑥2,𝑥≤1𝑥2+2𝑥,𝑥>1

28. 𝐺(𝑥) ={1/𝑥,𝑥<0𝑥,0≤𝑥

Find a formula for each function graphed in Exercises 29-32.

29. a.

b.

30. a.

b.

31. a.

b.

32. a.

b.

The Greatest and Least Integer Functions

33. For what values of 𝑥 is

a. ⌊𝑥⌋ =0?

b. ⌈𝑥⌉ =0?

34. What real numbers 𝑥 satisfy the equation ⌊𝑥⌋ =⌈𝑥⌉ ?

35. Does ⌈ −𝑥⌉ = −⌊𝑥⌋ for all real x? Give reasons for your answer.

36. Graph the function

Why is 𝑓(𝑥) called the integer part of 𝑥 ?

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

37. 𝑦 = −𝑥3

38. 𝑦 = −1𝑥2

39. 𝑦 = −1𝑥

40. 𝑦 =1|𝑥|

41. 𝑦 =√|𝑥|

42. 𝑦 =√−𝑥

43. 𝑦 =𝑥3/8

44. 𝑦 = −4√𝑥

45. 𝑦 = −𝑥3/2

46. 𝑦 =( −𝑥)2/3

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

47. 𝑓(𝑥) =3

48. 𝑓(𝑥) =𝑥−5

49. 𝑓(𝑥) =𝑥2 +1

50. 𝑓(𝑥) =𝑥2 +𝑥

51. 𝑔(𝑥) =𝑥3 +𝑥

52. 𝑔(𝑥) =𝑥4 +3𝑥2 −1

53. 𝑔(𝑥) =1𝑥2−1

54. 𝑔(𝑥) =𝑥𝑥2−1

55. ℎ(𝑡) =1𝑡−1

56. ℎ(𝑡) =|𝑡3|

57. ℎ(𝑡) =2𝑡 +1

58. ℎ(𝑡) =2|𝑡| +1

59. sin 2x

60. sin⁡𝑥2

61. cos 3x

62. 1 +cos⁡𝑥

Theory and Examples

63. The variable s is proportional to t, and s = 25 when t = 75. Determine t when s = 60.

64. Kinetic energy The kinetic energy K of a mass is proportional to the square of its velocity v. If K = 12,960 joules when v = 18 m/s, what is K when v = 10 m/s?

65. The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

66. Boyle’s law Boyle’s law says that the volume 𝑉 of a gas at constant temperature increases whenever the pressure 𝑃 decreases, so that 𝑉 and 𝑃 are inversely proportional. If 𝑃 =14.7 N/cm2 when 𝑉 =1000 cm3 , then what is 𝑉 when 𝑃 =23.4 N/cm2 ?

67. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 cm by 22 cm by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.

68. The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long.

a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

b. Express the area of the rectangle in terms of x.

In Exercises 69 and 70, match each equation with its graph. Do not use a graphing device, and give reasons for your answer.

T 71. a. Graph the functions 𝑓(𝑥) =𝑥/2 and 𝑔(𝑥) =1 +(4/𝑥) together to identify the values of 𝑥 for which

b. Confirm your findings in part (a) algebraically.

T 72. a. Graph the functions 𝑓(𝑥) =3/(𝑥 −1) and 𝑔(𝑥) =2/(𝑥 +1) together to identify the values of 𝑥 for which

b. Confirm your findings in part (a) algebraically.

73. For a curve to be symmetric about the x-axis, the point (𝑥,𝑦) must lie on the curve if and only if the point (𝑥, −𝑦) lies on the curve. Explain why a curve that is symmetric about the x-axis is not the graph of a function, unless the function is y = 0.

74. Three hundred books sell for 40𝑒𝑎𝑐ℎ,𝑟𝑒𝑠𝑢𝑙𝑡𝑖𝑛𝑔𝑖𝑛𝑎𝑟𝑒𝑣𝑒𝑛𝑢𝑒𝑜𝑓(300)(40) =12,000.𝐹𝑜𝑟𝑒𝑎𝑐ℎ5 increase in the price, 25 fewer books are sold. Write the revenue R as a function of the number x of $5 increases.

75. A pen in the shape of an isosceles right triangle with legs of length x m and hypotenuse of length h m is to be built. If fencing costs 5/𝑚𝑓𝑜𝑟𝑡ℎ𝑒𝑙𝑒𝑔𝑠𝑎𝑛𝑑10/m for the hypotenuse, write the total cost C of construction as a function of h.

76. Industrial costs A power plant sits next to a river where the river is 250 m wide. To lay a new cable from the plant to a location in the city 2 km downstream on the opposite side costs 180𝑝𝑒𝑟𝑚𝑒𝑡𝑒𝑟𝑎𝑐𝑟𝑜𝑠𝑠𝑡ℎ𝑒𝑟𝑖𝑣𝑒𝑟𝑎𝑛𝑑100 per meter along the land.

NOT TO SCALE

a. Suppose that the cable goes from the plant to a point Q on the opposite side that is x m from the point P directly opposite the plant. Write a function 𝐶(𝑥) that gives the cost of laying the cable in terms of the distance x.

b. Generate a table of values to determine if the least expensive location for point Q is less than 300 m or greater than 300 m from point P.

1.2 Combining Functions; Shifting and Scaling Graphs

In this section we look at the main ways functions are combined or transformed to form new functions.

Sums, Differences, Products, and Quotients

Like numbers, functions can be added, subtracted, multiplied, and divided (except where the denominator is zero) to produce new functions. If f and g are functions, then for every x that belongs to the domains of both f and g (that is, for 𝑥 ∈𝐷(𝑓) ∩𝐷(𝑔) ), we define functions 𝑓 +𝑔 , f - g, and fg by the formulas

Notice that the + sign on the left-hand side of the first equation represents the operation of addition of functions, whereas the + on the right-hand side of the equation means addition of the real numbers 𝑓(𝑥) and 𝑔(𝑥) .

At any point of 𝐷(𝑓) ∩𝐷(𝑔) at which 𝑔(𝑥) ≠0 , we can also define the function 𝑓/𝑔 by the formula

Functions can also be multiplied by constants: If 𝑐 is a real number, then the function 𝑐𝑓 is defined for all 𝑥 in the domain of 𝑓 by

EXAMPLE 1 The functions defined by the formulas

have domains 𝐷(𝑓) =[0,∞) and 𝐷(𝑔) =( −∞,1] . The points common to these domains are the points in

The following table summarizes the formulas and domains for the various algebraic combinations of the two functions. We also write 𝑓 ⋅𝑔 for the product function fg.