Study Note on Functions

By Gaurav Mohanty|Updated : December 13th, 2021

A functional relation connects two variable quantities x and y if each value of one quantity can be associated with one or several definite values of the other.

For example, y = x defines a relationship in which each point of x is related to a distinct point of y. These are: (1, 1), (2, 2) etc.

Similarly, y = 2x + 3; y = x2, y = x + 1/x … these are examples of functions.

We can write these as: y = f(x) = 2x + 3; y = f(x) = x2, y = f(x) = x + 1/x … these are examples of functions. The term f(x) denotes "function of x".

We write f(1) when we want the value of the function at the point when x = 1. Hence if f(x) = 5x + 4, then f(1) will be: 5(1) + 4 = 9 and f(2) will be: 5(2) + 4 = 14.

That is, we substitute the required value wherever x appears in the function.

Illustration 1: If f(x) = xx-2 find f(3).

Solution:

f(3) = 33-2 = 31 = 3.

• We should also be aware of the following notations:
• fog means f(g(x)).
• F(g(x)) means value of F when x = g(x).
• F(x).g(x) means F(x) × g(x)
• f.g means f(x) × g(x)
• f(a) means value of the function when x = a.

DOMAIN, RANGE AND INVERSE OF A FUNCTION

Real Function: Let A and B be two non empty sets. Then a function ‘f’ from set A to set B is a rule which associates elements of set A to elements of set B such that

1. all element of set A are associated to elements in set B.
2. an element of set A is associated to a unique element in set B.

If f is a function from a set A to set B, than we write f: A → B or A → B, meaning f is a function from A to B.

DOMAIN AND RANGE OF A FUNCTION

Let y = f(x) be a function.

The values that x takes are known as the domain, while the values that y takes is known as the range.

Illustration 3: Let f(x) = x2 and let x take the values in the set {–2, –1, 0, 1, 2}. Then f(x) can take only take the values 0, 1, 4. Hence Domain of (f) = {–2, –1, 0, 1, 2} and Range of (f) = B = {0, 1, 4}

INVERSE OF A FUNCTION

Illustration 4: If y = 2x, what is the inverse of the function?

Solution:

Here f(x) = 2x. Or x = y/2. Then f–1 = y/2.

Illustration 5: If f(x) = 2x + 7, find its inverse.

Solution:

y = 2x + 7, hence x = (y – 7)/2. Thus f–1 is written as (y – 7)/2

SOLVED EXAMPLES

Example 1: Find the domain and range of the function

Solution:

Clearly f(x) is not defined when x = 1. So domain (f) = R –{1}

Let . This shows that x is not a real number when y=–1,so range (f) = R – {–1}.

Example 2: Find the minimum value of the function f(x) = |x – 1| + |x – 2|.

Solution:

We note that both terms |x – 1| and |x – 2| can never be negative. Hence, the minimum value of each of terms is 0.

To obtain minimum value of the function, we put either of the terms as 0. Clearly, if x – 2 = 0, then the function attains minimum value of 1.

Example 3: Determine whether the function f(x) = x2 – |x| is even or odd:

Solution:

f(x) = x2 – |x| ⇒ f(–x) = (–x)2 – |–x| = x2 – |x| = f(x). Here f(- x) = f(x), so f(x) is an even function.

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