Section 1.1 Problems
1.2 Elementary Functions
1.2.1 What Is a Function?
Scientific investigations often study relationships between quantities, such as how enzyme activity depends on temperature or how the length of a fish is related to its age. To describe such relationships mathematically, the concept of a function is useful.
The wordfunction(or, more precisely, its Latin equivalentfunctio, which means
“execution”) was introduced by Leibniz in 1694 in order to describe curves. Later, Euler (1707–1783) used it to describe any equation involving variables and constants.
The modern definition is much broader and emphasizes the basic idea of expressing relationships between any two sets.
Definition Afunction f is a rule that assigns each element x in the set A exactly one elementyin the setB. The elementyis called theimage(orvalue) ofx under f and is denoted by f(x)(read “f ofx”). The set Ais called the domainof f, the setBis called thecodomainof f, and the set f(A) = {y : y= f(x)for somex ∈A}is called therangeof f.
To define a function, we use the notation f : A→ B
x→ f(x)
where Aand Bare subsets of the set of real numbers. Frequently, we simply write y= f(x)and callxtheindependentvariable andythedependentvariable. We can illustrate functions graphically in the x–y plane. In Figure 1.9, we see the graph of y= f(x), with domain A, codomainB, and range f(A).
a y
A B
b x f(b)
f(a) f(A)
Figure 1.9 A function f(x)with domainA, codomainB, and range
f(A).
The function f(x)must be specified; for example, f(x)could be given by a graph as in Figure 1.9, or it could be expressed algebraically, such as f(x)=x2. Note that f(A)⊂B, but not every element in the codomainBmust be in f(A). For instance, let
f : R→R x →x2
The domain of f isR, but the range of f is only[0,∞)because the square of a real number is nonnegative; that is, f(R)= [0,∞)=R. Also, the domain of a function need not be the largest possible set on which we can define the function, asRis in the preceding example. For instance, we could have defined f on a smaller set, such as[0,1], calling the new functiong, given by
g: [0,1] →R x →x2
Although the same rule is used for f and g, the two functions are not the same, because their respective domains are different.
Two functions f andgareequalif and only if 1. f andgare defined on the same domain, and 2. f(x)=g(x)for allxin the domain.
EXAMPLE 1 Let
f1: [0,1] →R x →x2 f2: [0,1] →R
x →√ x4
and
f3: R→R x→x2 Determine which of these functions are equal.
Solution
Because f1and f2 are defined on the same domain and f1(x)= f2(x)= x2for all x ∈ [0,1], it follows that f1and f2are equal.Neither f1 nor f2is equal to f3, because the domain of f3 is different from the domains of f1and f2.
The choices of domains for the functions that we have thus far considered may
x y
y f(x)
Figure 1.10 The vertical line test shows that the graph ofy= f(x)is a function.
look somewhat arbitrary (and they are arbitrary in the examples we have seen so far).
In applications, however, there is often a natural choice of domain. For instance, if we look at a certain plant response (such as total biomass or the ratio of above to below biomass) as a function of nitrogen concentration in the soil, then, given that nitrogen concentration cannot be negative, the domain for this function could be the set of nonnegative real numbers. As another example, suppose we define a function that depends on the fraction of a population infected with a certain virus; then a natural choice for the domain of this function would be the interval[0,1]because a fraction of a population must be a number between 0 and 1.
In our definition of a function, we stated that a function is a rule that assigns, to each elementx ∈ A,exactlyone elementy ∈ B. When we graph y = f(x)in the x–yplane, there is a simple test to decide whether or not f(x)is a function: If each vertical line intersects the graph ofy = f(x)at most once, then f(x)is a function.
Figure 1.10 shows the graph of a function: Each vertical line intersects the graph of y= f(x)at most once. The graph ofy= f(x)in Figure 1.11 is not a function, since
x
y y f(x)
Figure 1.11 The vertical line test shows that the graph ofy= f(x)is not a function.
there arex-values that are assigned to more than one y-value, as illustrated by the vertical line that intersects the graph more than once.
Sometimes functions show certain symmetries. For example, in Figure 1.12, f(x) = x is symmetric about the origin; that is, f(x) = −f(−x). In Figure 1.13, g(x)= x2is symmetric about the y-axis; that is,g(x)=g(−x). In the first case, we say that f is odd; in the second case, thatgis even. To check whether a function is even or odd, we use the following definition:
A function f : A→Bis called
1. evenif f(x)= f(−x)for allx∈ A, and 2. oddif f(x)= −f(−x)for allx∈ A.
1
2 1 1 0
2 3
4 3 2 1 2 3 4 5
3
f(x) x y
x
Figure 1.12 The graph ofy=xis symmetric about the origin.
10 8 6 4 2
3 2 1 0 1 2 3 x y
g(x) x2
Figure 1.13 The graph ofy=x2is symmetric about the y-axis.
Using this criterion, we can show that f(x)= x,x∈R, is an odd function:
−f(−x)= −(−x)=x= f(x) for allx ∈R Likewise, to show thatg(x)=x2,x ∈R, is an even function, we compute
g(−x)=(−x)2=x2=g(x) for allx∈R
We will now look at the case where one quantity is given as a function of another quantity that, in turn, can be written as a function of yet another quantity. To illus- trate this situation, suppose we are interested in the abundance of a predator, which depends on the abundance of a herbivore, which, in turn, depends on the abundance of plant biomass. If we denote the plant biomass byx and the herbivore biomass byu, thenx anduare related via a functiong, namely,u = g(x). Likewise, if we denote the predator biomass byy, thenuandyare related via a function f, namely, y = f(u). We can express the predator biomass as a function of the plant biomass by substitutingg(x)foru. That is, we findy = f[g(x)]. Functions that are defined in such a way are called composite functions.
Definition Thecomposite function f ◦g (also called thecompositionof f andg) is defined as
(f ◦g)(x)= f[g(x)]
for eachxin the domain ofgfor whichg(x)is in the domain of f.
The composition of functions is illustrated in Figure 1.14. We callgthe inner function and f the outer function. The phrase “for eachxin the domain ofgfor whichg(x)is in the domain of f” is best explained with the use of Figure 1.14. In order to compute f(u),uneeds to be in the domain of f. But sinceu = g(x), we really require that f[g(x)]
g(x) x
g f
Figure 1.14 The composition of functions.
g(x)be in the domain of f for the values ofxwe use to computeg(x). EXAMPLE 2 If f(x)=√
x,x ≥0, andg(x)=x2+1,x ∈R, find (a) (f ◦g)(x)and (b) (g◦ f)(x).
Solution
(a) To find(f ◦g)(x), we set f(u)=√uandg(x)= x2+1. Then y= f(u)= f[g(x)] = f(x2+1)=
x2+1
To determine the domain of f ◦g, we observe that the domain of the inner function gisRand its range is[1,∞). Since the range ofgis contained in the domain of the outer function f ([1,∞)⊂ [0,∞)), the domain of f ◦gisR.
(b) To find(g◦ f)(x), we setg(u)=u2+1 and f(x)=√
x. Then y=g(u)=g[f(x)] =g(√
x)=(√
x)2+1=x+1
To determine the domain ofg◦ f, we observe that the domain of the inner function f is[0,∞)and its range is[0,∞). The range of f is contained in the domain of the outer functiong([0,∞)⊂R), so the domain ofg◦ f is[0,∞).
In the last example, you should observe that f ◦gis different fromg◦ f, which implies that the order in which you compose functions is important. The notation f ◦gmeans that you applygfirst and then f. In addition, you should pay attention to the domains of composite functions. In the next example, the domain is harder to find.
EXAMPLE 3 If f(x)=2x2,x ≥2, andg(x)=√
x,x ≥0, find(f◦g)(x)together with its domain.
Solution
We compute(f ◦g)(x)= f[g(x)] = f(√
x)=2(√
x)2=2x
This part was not difficult. However, finding the domain of f◦gis more complicated.
The domain of the inner functiongis the interval[0,∞); hence, the range ofgis the
g(x)
0 4
Restricted domain of f ⴰ g
Domain of g(x)
0 2
Domain of f(x)
Range of g(x)
Figure 1.15 Finding the domain of a compositie function: The domain of g(x)must be restricted in Example 3.
interval[0,∞). The domain of f is only[2,∞), which means that the range ofgis notcontained in the domain of f. We therefore need to restrict the domain ofgto ensure that its range is contained in the domain of f. We can choose only values of x such thatg(x)∈ [2,∞). Sinceg(x) = √
x, we need to restrictxto[4,∞). Thus, for everyx∈ [4,∞),g(x)∈ [2,∞), which is the domain of f. Therefore,
(f ◦g)(x)=2x, x ≥4 See Figure 1.15.
In the subsections that follow, we introduce the basic functions that are used throughout the remainder of this book.