3 Functions
3.1. The Notion of a Function We already have a general notion of a function, since we have drawn straight line, quadratic and other curves.
y
x
Figure 3.1 A graph is a diagram which represents a rule. To every value a, the rule assigns a new number, b, say. As a varies, so does the value of b. The graph is the collection of all points (a, b), where each b has been determined from a by the conversion rule. To indicate that b depends on a, we often write b = f (a) to indicate that a conversion rule, which we are calling f , has acted on a to obtain b. The fact that this collection of points often forms a smooth curve is a reflection of the type of conversion rule; it is not always the case, since it possible to write down some pretty wild conversion rules! If we know the specific form of the conversion rule, and have an explicit formula which tells us what f (a) actually is, then the graph is unnecessary, since all the information contained in the graph is described by that formula. Thus, rather than consider the quadratic graph y = x2 −3x+1, we could consider the function f (x) = x2 − 3x + 1. For Interest Instead of writing f (x) = x2 − 3x + 1, some authors write f : x 7→ x2 − 3x + 1, which can be read as “the function f such that x maps to x2 − 3x + 1. There is no difference in meaning between these two notations.
A function can be thought of as a “black box" — if you put in an input value x, out comes a definite output value. If we decide to label this particular function/“black box" by calling it f , then the output value is f (x). The output value f (x) is called the image of x under f . 34
3.2. The Domain and the Range
35
Going back over the last two Chapters, we observe that polynomials and rational functions (sic ) are all functions, and we have been using the f (x) notation to represent them. Here are some examples of functions. Some of them can be defined simply in algebra, others take more effort; the key thing is that each function assigns a unique output value to each input value. Note that we are free to give functions any name we like — they don’t all have to be called f ! 3 4 3 πx , 1 2 (x + |x|),
f (x)
=
g(x)
=
a(x)
=
k(x)
=
m(x)
=
p(n)
=
the nth prime number, for any positive integer n,
π(x)
=
q(x)
=
the number of prime numbers ≤ x, √ x,
d(n)
=
the number of positive integer divisors of the positive integer n,
H(n)
=
1 + 21 + 13 + 14 + · · · + n1 , for any positive integer n.
1 , x−1 √ 1 , x2 +1 { 1, 0,
x∈Q, x<Q,
3.2. The Domain and the Range Looking at the examples of functions given above, we note different functions have different properties. To begin with, not all functions can accept all inputs. For example, the function a(x) makes no sense when x = 1, while the function q(x) makes no sense unless x ≥ 0. Even worse, p(n), d(n) and H(n) only accept inputs which are positive integers. To understand a function, we need to know which numbers it accepts as inputs. Key Fact 3.1 Domain and Range The set of all possible input x values for a function f is called the domain of f , and is denoted Dom(f ). The set of all possible images of f is called the range of the function, and is denoted Ran(f ).
A function has not been properly defined until both its rule f (x) and its domain have been specified. We shall meet a more formal notation for functions shortly, but the following shorthand notation is adequate for most purposes. Here are examples of functions: f1 (x) = 43 πx3 , f2 (x) = a1 (x) = a2 (x) =
4 3 3 πx 1 x−1 , 1 x−1 ,
,
√
x∈R, x ∈ R, x > 0, x ∈ R, x , 0, x ∈ R, x > 0,
q(x) = x ,
x ∈ R, x ≥ 0,
d(n) = the number of positive integer divisors of n ,
n ∈ N,
s(n) = the sum of the digits in the decimal expansion of n ,
n ∈ N , n ≥ 10 .
Note that it is possible for different functions to have the same rule applied to different domains. We have already met this idea when considering problems where physical considerations might lead us to accept certain solutions and discard others. Suppose that one side of a rectangle is 4 cm shorter than the other side, and that the area of the rectangle is 21 cm2 . Calling the two sides
3.2. The Domain and the Range
36
x cm and x − 4 cm, the area of the rectangle is x(x − 4) cm2 . Solving the equation x(x − 4) = 21, we deduce that x = 7 or −3. Since we cannot have a rectangle of sides −3 cm and −7 cm, we discard the x = −3 solution and deduce that x = 7, so that the rectangle is 7 cm by 3 cm. Using the ideas of functions, we can say instead that the area function is A(x) = x(x − 4) ,
x ∈ R, x > 4 ,
and then the only value of x in the domain for which A(x) = 21 is x = 7. Defining the correct domain for the area function ruled out having to consider rectangles with negative edge lengths. In short, a function is specified by first defining its rule, then adding a collection of conditions which are sufficient to specify the domain. Technically, the domains of a2 and s are { } { }
x ∈ R
x > 0 and n ∈ N
n ≥ 10 since the domains are sets, not conditions, but this abuse of notation is convenient and easy to use. Thus a domain might be written down as a set (for example, R or Q), or else as a collection of conditions (for example, x ∈ R , x > 0 or x ∈ N , 1 ≤ n ≤ 100). Similarly, a range might be written as a set (such as R), or else as a set of conditions (such as x ∈ R , x > 0). If we know the name of the function, the conditions might be written down explicitly involving that name (such as f (x) ∈ R , f (x) ≥ 2). √ Example 3.2.1. If f (x) = x + 1,
(a) what is the largest possible subset of the reals R that can be a domain for f , and what is its corresponding range, (b) what is the range when the domain is x ∈ R , 3 ≤ x ≤ 8, (c) which integers are in the range of f if the domain is N? (a) We cannot take the square root of a negative number, and so the largest possible domain is x ∈ R , x ≥ −1 (to ensure that x + 1 ≥ 0 throughout the domain). Clearly f (x) ≥ 0 for all x ≥ −1, and since y 2 − 1 is in the domain of f , with f (y 2 − 1) = y, for any y ∈ R , y ≥ 0, we see that any nonnegative real number is in the range. The range of f is f (x) ∈ R , f (x) ≥ 0. (b) If x ∈ R with 3 ≤ x ≤ 8 then 2 ≤ f (x) ≤ 3. On the other hand, if y ∈ R with 2 ≤ y ≤ 3, then y 2 − 1 belongs to the domain of f and f (y 2 − 1) = x, and so x is in the range of f . Thus the range of f is f (x) ∈ R , 2 ≤ f (x) ≤ 3. √ (c) Since f (n) = n + 1 for any positive integer n, the range of f{ consists } numbers of the
of all √ √ form n + 1, where n is a positive integer. Thus the range is n + 1 n ∈ N . Another way of {√
} describing this set is n
n ∈ N , n ≥ 2 . √ Since n = n2 for any n ∈ N, and n2 ≥ 2 for any positive integer n except 1, we see that the range contains all positive integers except 1.
Exercise 3A
1. Given f (x) = 2x + 5, find the values of
a) f (3),
b) f (0),
c) f (−4),
) ( d) f − 21 .
c) f (3),
d) f (3).
c) f (1),
d) f (3).
2. Given f (x) = 3x2 + 2, find the values of
a) f (4),
b) f (1),
3. Given f (x) = x2 + 4x + 3, find the values of ( ) a) f (2), b) f 21 ,
3.3. Graph Transformations
37
4. Given g(x) = x3 and h(x) = 4x + 1,
a) find the value of g(2) + h(2); c) show that g(5) = h(31);
b) find the value of 3g(−1) − 4h(−1); d) find the value of h(g(2)).
5. Given f (x) = xn and f (3) = 81, determine the value of n. 6. Given that f (x) = ax + b and that f (2) = 7 and f (3) = 12, find a and b. 7. Find the largest possible subset of R which can be a domain of each of the following functions. √ √ √ √ a) x b) −x c) √x − 4 d) √4 − x √ √ e) x(x − 4) f) 2x(x − 4) g) x2 − 7x + 12 h) x3 − 8 1 1 1 1 i) x−2 j) √ k) 1+√x l) (x−1)(x−2) x−2
8. The domains of these functions are the set of all positive real numbers. Find their ranges.
b) f (x) = −5x e) f (x) = (x + 2)2 − 1
a) f (x) = 2x + 7 d) f (x) = x2 − 1
c) f (x) = 3x − 1 f) f (x) = (x − 1)2 + 2
9. Find the range of each of the following functions. All the functions are defined for all real values of x.
a) f (x) = x2 + 4 d) f (x) = −(1 − x)2 + 7
b) f (x) = 2(x2 + 5) e) f (x) = 3(x + 5)2 + 2
c) f (x) = (x − 1)2 + 6 f) f (x) = 2(x + 2)4 − 1
10. These functions are each defined for the given domain. Find their ranges.
a) f (x) = 2x for x ∈ R , 0 ≤ x ≤ 8 c) f (x) = x2 for x ∈ R , −1 ≤ x ≤ 4
b) f (x) = 3 − 2x for x ∈ R , −2 ≤ x ≤ 2 d) f (x) = x2 for x ∈ R , −5 ≤ x ≤ −2
11. Find the range of each of the following functions. All the functions are defined for the largest possible domain of values of x.
a) f (x) = x8 e) f (x) = x4 + 5
b) f (x) = x11 f) f (x) = 14 x + 18
c) f (x) = x13 √ g) f (x) = 4 − x2
d) f (x) = x14 √ h) f (x) = 4 − x
12. A piece of wire 24 cm long has the shape of a rectangle. Given that the width is w cm, show that the area, A cm2 , of the rectangle is given by the function A = 36 − (6 − w)2 . Find the greatest possible domain for the area function in this context. 13. Given that a cuboid has height x cm, length (22 − 2x) cm and width (8 − 2x) cm, state an appropriate domain for the function representing the volume of the cuboid.
3.3. Graph Transformations The method shown above for determining the range of a function is precise, but laborious! In many cases, a graphical approach will suffice, and so it will be useful to add to the techniques we already know for graph-sketching. There are a variety of curves whose shapes we already know, such as straight lines, quadratics, cubics. We also know the shape of reciprocal graphs, such as y = 1x , and the graph of the modulus function y = |x|.
3.3. Graph Transformations
38 y
y
y
x
x
x
y=x
y
y
= x2
y = x3 y
y
x
x
x
y = x3 − 6x y = 1/x y = |x| Figure 3.2 We have already discussed methods for sketching more general quadratic graphs, looking for intercepts with the axes and identifying the vertex, but similar methods for cubics are that much more complex. Factorising cubics is not straightforward, and identifying whether or not a particular cubic has local maxima or minima, as in graph (d) above, is not easy to do until we have studied more calculus. It is useful, therefore, to have an array of geometric techniques which enable us to sketch more curves than the standard six above relatively easily. This will be achieved by recognising how certain algebraic operations on the function are reflected by some standard geometric transformations to their graphs. • Translations Consider the three equations y = x2
,
y = x2 + 3
y = x2 − 2.
,
If we introduce the function f (x) = x2 , then these three equations are y = f (x)
,
y = f (x) + 3
,
y = f (x) − 2.
and so the second and third equations have been derived from the first by adding or subtracting a constant to the function f (x). If we compare the graphs of these functions, then we see that the graph of the second function has been obtained by translating the graph of the first function a distance of 3 units parallel to the y-axis, while the graph of the third function has been obtained by translating the graph of the first function a distance of −2 units parallel to the y-axis. Here a negative translation is a translation “down", in the direction of the negative y-axis.
y
y=x2 +3
y=x2 x y=x2 -2
Figure 3.3
Key Fact 3.2 Graph Transformations: Vertical Translations The graph of the equation y = f (x) + a is obtained from the graph of the equation y = f (x) by translating it through a distance of a units parallel to the y-axis.
This transformation is easy to understand — adding a to f (x) increases the value of the function by a, and hence lifts its graph a height of a units. It can also be described in vector ( ) terms as translation by the vector 0a .
3.3. Graph Transformations
39
How to perform a horizontal translation through a units is less clear. Consider the following graph of y = x2 + 1 and y = (x − a)2 + 1.
y = x2 +1
y = (x-a)2 +1 a
Figure 3.4
It certainly looks as if the graph of y = (x − a)2 + 1 has been obtained from the graph of y = x2 + 1 by translating it through a distance of a units parallel to the x-axis. Why does this work? Consider the equations y = f (x) and y = f (x−a). If we introduce a new coordinate X = x−a, then the second equation becomes y = f (X), and hence the graph of y = f (x − a) will look exactly the same, plotted on the Xy-axis system, as the graph of y = f (x) when plotted on the xy-axis system. The question that remains is to find what the Xy-axis system looks like, compared to the xy-axis system. A point with coordinates (x, y) in the xy-axis system has coordinates (x − a, y) in the Xy-axis system, and so the point (a, 0) in the xy-axis system becomes the origin of the new Xy-coordinate system. It is clear that the Xy-axis system is obtained by translating the xy-axis system a distance of a units parallel to the x-axis: y y a
X
x
Figure 3.5 and so we have:
Key Fact 3.3 Graph Transformations: Horizontal Translations The graph of the equation y = f (x − a) is obtained from the graph of the equation y = f (x) by translating it through a distance of a units parallel to the x-axis. This ( ) transformation can also be described as translation by the vector 0a .