Functions and Their Graphs
CHAPTER 4 : 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.
4.1.1 Notation
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)
4.2 Graphs 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
4.2.2 Graph of a Linear Function
a. A linear function is defined as , i.e. the highest power of x is 1. The common way to
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.
4.1.1 Notation
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)
4.2 Graphs 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
4.2.2 Graph of a Linear Function
a. A linear function is defined as , i.e. the highest power of x is 1. The common way to