Topic Functions

LEARNING OUTCOMES
By the end of this topic, you should be able to: 1. 2. 3. 4. 5. Distinguish function and non function; Sketch the graph a function; Represent a function algebraically; Visualise the translation and shifting of a function; and Determine the symmetry properties of a function.

INTRODUCTION
Welcome to the course in calculus. This branch of mathematics has its origin from the study of the basic physical properties of our universe, such as the motion of cars, planets, and molecules. There are two divisions of calculus, namely differential and integral calculus. Differential calculus describes the methods by which, given a function, we can find its associated rate of change, called the derivative. The function may describe a constantly changing system, such as the temperature variation over the course of the day or the velocity of a planet around a star over the course of one rotation. The derivative of those functions would give us the rate of change of the temperature or the acceleration of the planet respectively.

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Integral calculus on the other hand is the opposite of differential calculus. Thus, given the rate of change in a system, we can find the values that describe the input into the system. Suppose we are given the derivative, like acceleration, we can use the anti-derivative or integration to find the original function, say velocity. We may also use integration to calculate values such as the area under a curve, the surface area, or the volume of a solid. This is possible since we begin by approximating an area with a series of rectangles, and make our guess more accurate by studying the limiting value. The limit, or the number towards which the approximations tend, will give us the precise result. The fundamental principle in calculus relies on the fact that we can always use approximations of increasing accuracy to obtain a precise solution to a problem. For instance, we can approximate a curve by a series of short straight lines or a spherical solid by a series of small cubes that fit inside the sphere in each iteration. Using calculus, we can determine the approximations tend to a precise result, called limit, which will accurately describe and reproduce the curve, surface, or solid under study. We may use calculus to help solve problems in mathematics, physics, engineering and other applied technology. In this topic we shall be studying the fundamental object in calculus: function. We shall be discussing the basic concepts of function, its graph, and the various ways of function transformations.

1.1
1.1.1

FUNCTION AND ITS REPRESENTATION

You need to be familiar and be able to recognise the various types of equations and their graphs before you proceed further. Let us review the following equations: (a)

2 y = x +1 3
x2 = y

equation of a straight line, equation of a parabola.

(b)

If we substitute a certain value of x in equation (a) or (b), we will notice that y will have a certain value. In addition to that, each value of y in case (b) can be generated from two distinct values of x. You may do some substitutions for distinct values of x and verify the results.

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From the results of the two cases (a) and (b) above, we are able to recognise two types of relations that associate independent variable x with the dependent variable y, they are: (i) (ii) 2 One to one such as in equation y = x + 1; and 3 Many to one such as in equation x2 = y.

The relations one to one and many to one are called function. However, there is an association that relates x with y by the relation one to many. This latter relation does not form a function. Study the following examples to illustrate our discussion.
Example 1.1 Consider the equation of a circle x2 + y2 = 25 for all real values of x and y. Determine the relation between x and y. Solution If x = 4 is substituted into the above equation, we obtain y = 3 or y = 3. Observe that we have the relation which associates one value of x to two differenrt values of y. Thus, the relation between x and y is one to many.

Example 1.2 Consider the equation of a parabola y2 = x for all real values of x and y. Determine the relation between x and y. Solution By substituting x = 4, we obtain y = 2 or y = 2. Hence, a single value of x is associated with two distinct values of y. Therefore we have the relation between x and y as one to many.

The results in Example 1.1 and Example 1.2 indicate that the value of y that is associated with a particular value of x is not unique. Thus the equations x2 + y2 = 25 and y2 = x do not meet the criteria of a function.

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As a mathematical convention, we write a function in a special notation. For 2 example, the equation y = x + 1 can be written in the form y = f ( x) where the 3 function f is given by

2 f ( x) = x + 1. 3 Such notation was introduced by Leonhard Euler in 1734 and the word function was first used by Gottfried Wilhelm Leibniz in 1694.
SELF-CHECK 1.1

Determine whether each of the following diagrams represents graph of a function or an equation (a) (b)

(c)

(d)

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1.1.2

Graph of a Function

The graph of a function f is a set of points {(x, f ( x)) where x is in the domain of f }.

Figure 1.1: Graph of y 2 = x

Using the vertical line test, line AB cuts the curve that represents the graph of the equation y2 = x at two points, A and B as shown. This shows that the graph drawn does not represent a function. The graphs in Figure 1.2(a), (b), (c) and (d) represent functions, namely a constant function, linear function, modulus function and a piecewise function respectively. You may use the vertical line test to confirm that the graphs in Figure 1.2(a), (b), (c) and (d) represent functions.

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Figure 1.2(a), (b), (c) & (d): Graphs of functions

Observe that Figure 1.2(d) represents graph of a function 1 , f ( x) = 1 , x>0 x<0 x , x0 x

which is usually written as f ( x) =

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Example 1.3

Sketch the graph of f ( x) =

1 . x2

Solution Firstly, observe that the function is undefined at x = 0. We also observe that x can be any value, positive or negative and thus we conclude that the function is defined for all values of x except for x = 0.

Secondly, since x 2 > 0, the value of f ( x) = Hence, the graph of f ( x) =

1 is always positive. x2

1 is always above the x-axis. x2

1 tends to be a very small x2 1 1 value, i.e. f ( x) = 2 0. Similarly, as x , f ( x) = 2 0. Thus we x x 1 say that the line y = 0 or the x-axis is an asymptote of the graph of f ( x) = 2 . x Next, we observe as x , the value of f ( x) =
Finally, as x tends to zero from the right (written as x 0+ ), the value of 1 1 f ( x) = 2 tends to be very large, i.e. f ( x) = 2 . Similarly, as x tends to x x 1 zero from the left (written as x 0 ), the value of f ( x) = 2 tends to be very x 1 large, i.e. f ( x) = 2 . Therefore the line x = 0 or the y-axis is also an x 1 asymptote of the graph f ( x) = 2 . We therefore obtain the sketch of the x 1 graph of f ( x) = 2 as in Figure 1.3. x

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Figure 1.3: Graph of f ( x) =

1 x2

Example 1.4

Sketch the graph of g ( x) =

1 . x

Solution First observe that the function is undefined at x = 0. We also observe that x can be any value, positive or negative and thus we conclude that the function is defined for all values of x except at x = 0. Furthermore, when x is negative, the 1 value of the function g ( x) = is also negative and when x is positive, the x 1 value of the function g ( x) = is also positive. x

Therefore, we conclude that the function g ( x) =

1 is always negative for all x negative values of x and it is always positive for all positive values of x. Next, we observe as x , the value of g ( x) =

1 tends to a very small x 1 1 value, i.e. g ( x) = 0. Similarly, as x , g ( x) = 0. Thus we say x x 1 that the line y = 0 or the x-axis is an asymptote of the graph of g ( x) = . x

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Finally, as x tends to zero from the right (written as x 0+ ), the value of 1 1 g ( x) = tends to be very large and positive, i.e. g ( x) = . Similarly, as x x 1 x tends to zero from the left (written as x 0 ), the value of g ( x) = tends x 1 to be very large but negative, i.e. g ( x) = . Therefore the line x = 0 or x 1 the y-axis is also an asymptote of the graph g ( x) = . We therefore obtain the x 1 sketch of the graph of g ( x) = as in Figure 1.4 below. x

Figure 1.4: Graph of g ( x) =

1 x

SELF-CHECK 1.2

Can you name other examples of graphs of functions?

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SELF-CHECK 1.3

Sketch the graphs of f ( x) = x x and g ( x) = x 2 x .

1.2

SYMMETRY PROPERTY OF FUNCTION

We have already been able to determine the types of graphs of functions and nonfunctions. Next, we shall study the symmetry property of a graph. This will then enable us to determine the shape of a graph on the other side of an axis. We shall now define the symmetry properties of a graph.
Symmetry Property 1 The graph of the equation y = f ( x) is said to be symmetrical about the y-axis if the positive and negative values of x result in the same value of y, that is f ( x) = f ( x).

Graphically, we can illustrate the symmetry property 1 as in Figure 1.5 below.

Figure 1.5: Symmetry about the y-axis

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Example 1.5 Based on the symmetry property 1, sketch and obtain the symmetry of the two graphs of y = x2 and y = cos x. Such function y is said to be an even function of the variable x.

Symmetry Property 2 The graph of the equation y = g(x) is said to be symmetrical about the x-axis if the positive and negative values of y result in the same value of x, that is g(y) = g(y).

Graphically, the symmetry property 2 is illustrated in Figure 1.6 below.

Figure 1.6: Symmetry about the x-axis

Example 1.6 Based on the symmetry property 2, sketch the graph of the equation x = y2. Such symmetric property implies that y is not a function of the variable x.

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Symmetry Property 3 The graph of the equation y = f ( x) is said to be symmetrical about the origin if any change in the sign of the value of x results in the change of the sign of the value of y.

Graphically, symmetry about the origin is illustrated in Figure 1.7 below.

Figure 1.7: Symmetry about the origin

Example 1.7 If f ( x) = x3, we obtain f (x) = x3 = f ( x). The graph of this function is symmetrical about the origin. Explain, why? Sketch the graph of this function and observe the shape.

1.3

TRANSLATION OF GRAPH

We have already studied symmetry properties of graph. In this section we shall be concerned about the translation of a graph. Next, we shall consider the changes or manipulations that can be made on a graph. We may shift a graph up and down or left and right.

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Now, let us study about the translation of a graph. Suppose the graph of an equation y = f ( x) is already drawn, can we visualise the graph of y = f ( x) + c and y = f ( x) c assuming that c is a positive constant? Study the following examples.
Example 1.8 Consider the graph of y = x2. Sketch the graphs of y = x2 and y = x2 2. Solution The graph of y = x2 is shown in Figure 1.8 by the thick line. While the graph of y = x2 2 is obtained by shifting downward the graph of y = x2 an amount of 2 units. This is illustrated in Figure 1.8 by the dotted line.

Figure 1.8: Graph of y = x2 and y = x2 2

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SELF-CHECK 1.4

Sketch the graphs of y = x2 and y = x2 + 2 on the same axes. From the graph of y = f ( x) we can also obtain the graph of y = f ( x + c) and y = f ( x c). The following example illustrates our discussion.
Example 1.9 Obtain the graph of y = (x + 2)2 from the graph of y = x2. Solution The graph of y = x2 + 2 is the graph of y = x2 shifted to the left by an amount of 2 units. This is given in Figure 1.9 below. The graph of y = (x + 2)2 is represented by the dotted line and the graph of y = x2 is represented by the thick line.

Figure 1.9: Graphs of y = x2 and y = (x + 2)2

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SELF-CHECK 1.5

Try to obtain the graph of y = (x 2)2 from the graph of y = x2.

SELF-CHECK 1.6

1.

Determine the symmetry property of each of the following graphs.

2.

Based on the graph of y = x2, sketch the graph of y = ( x 2) x 2 + 1.

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In Topic 1 we have learnt about function and its graph. From the graph of a function, we are able to determine the symmetry property and the translation and shifting of a graph of a function. We may summarise as follows:
Equation Property

Function (i) (ii) 2 y = x +1 3 2 x =y One to one relation. Many to many relation.

Non Function x 2 + y 2 = 25 (Circle) y 2 = x (Parabola) Symmetry Property 1 f ( x) = f ( x)

One to many relation. One to many relation. y = f ( x) is said to be symmetrical about the y-axis if the positive and negative values of x result in the same value of y. y = g(x) is said to be symmetrical about the x-axis if the positive and negative values of y result in the same value of x. y = f ( x) is said to be symmetrical about the origin if any change in the sign of the value of x result in the change of the sign of the value of y. Shifting upward Shifting downward Shifting to the left Shifting to the right

Symmetry Property 2 g(y) = g(y) Symmetry Property 3 f ( x) = y, f ( x) = y

Translation y = x2 + 2 y = x2 2 y = (x + 2)2 y = (x 2)2