# Functions and Their Graphs

4.1 Definition of Function

A function from one set X to another set Y is a rule that assigns each element in X to one element in Y.

4.1.1Notation

If f denotes a function from X to Y, we write

4.1.2 Domain and range

X is known as the domain of f and Y the range of f. (Note that domain and range are sets.)

4.1.3 Object and image

If and , then x and y are known respectively as the objects and images of f. We can write

, , .

We can represent a function in its general form, that is

f(x) = y.

Example 4.1

a. Given that , find f(0), f(1) and f(2).

Example 4.2

a. Given that , find the possible values of a such that

(a)f(a) = 4,(b)f(a) = a.

Solution

a. Given that , find f(0), f(1) and f(2).

b. Given that , find the possible values of a such that

(a)f(a) = 4,(b)f(a) = a.

(a)

(b)

4.2Graphs of Functions

An equation in x and y defines a function y = f(x) if for each value of x there is only one value of y.

Example:

y = 3x +1,,.

The graph of a function in the x-y plane is the set of all points (x, y) where x is the domain of f and y is the range of f.

Example

Figure 1 below shows the graph of a linear function, the square root function and a general function.

y = f(x)

y = x

(a) (b) (c)

Figure 1

It is easy to read the domain and range from the graph of a function, as shown in Figure 2.

Figure 2

4.2.1 Vertical Line Test

Vertical line test is usually applied to a graph to determine if it represents a function. If a vertical line intersects the graph at only one point, then each number x determines exactly one value of y and the graph represents a function y = f(x).

A function Not a function

4.2.2Graph of a Linear Function

a.A linear function is defined as

,

i.e. the highest power of x is 1.

The common way to write a linear function is , .

b.The graph of a linear function is a straight line.

c.The slope (gradient) of the line depends on the sign of a:

(i), the line “slopes up”,

(ii), the line “slopes down”.

d.f(0) = b i.e. b is the intercept on the y-axis.

4.2.3Graph of a quadratic function

a. A quadratic function is defined as

,

or more simply

,

i.e. the highest power of x is 2.

b. graph of a quadratic function is U-shaped (i.e. a parabola) which depends on the sign of a.

(i)If a > 0, the curve “concaves down”.U

(ii)If a < 0, the curve “concaves up”.U

c. The vertex of the parabola is , where .

d. To sketch the graph of a quadratic function we make use of:

(i) the sign of a i.e. whether the parabola is concaved down or concaved up,

(ii) the value f(0), i.e. where it cuts the y-axis,

(iii)the vertex .

4.3Graph of a Split Function

A split function is a function defined over two or more than two intervals.

Examples

(a)(b)

(c)

4.4Composite Function

4.4.1f and g are two functions. The composite function gf is a combination of f and g such that the range of f is the domain of g.

4.4.2Generally .

4.4.3 and

4.5 Inverse Function

4.5.1Definition:

If f and g are two functions of x such that

or

then f is the inverse of g and g is the inverse of f.

4.5.2Notation:

The inverse of a function f is written as . We have, from 4.4.1,

or

.

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