Functions are a fundamental topic in algebra, taught in Classes 11 and 12. This guide will explain key concepts related to functions in algebra, provide solved examples, and offer function practice questions to help you excel in exams. Here, you’ll gain the skills needed to solve various functions questions with confidence.

What is a Function?

A function is a fundamental concept in mathematics, particularly in algebra, where it describes a relationship between a set of inputs and a set of permissible outputs. Each input is related to exactly one output. The set of inputs is called the domain, and the set of outputs is called the codomain.

In more formal terms, a function f from a set A (the domain) to a set B (the codomain) is a rule that assigns each element x in A exactly one element y in B. This relationship is often written as f: A→B, and f(x) = y.

Types of Functions

• One-to-One Function (Injective): Each element of the domain is mapped to a unique element in the codomain.
• Onto Function (Surjective): Every element of the codomain is mapped by at least one element of the domain.
• Bijective Function: A function that is both one-to-one and onto, meaning there is a perfect pairing between the domain and codomain elements.
• Constant Function: A function that always returns the same value regardless of the input.
• Identity Function: A function that returns the input as the output, f(x)=x.

Important Formulas of Functions

Some of the important formulas required to solve questions of functions are mentioned below:

(f + g)(x) = f(x) + g(x)

(f − g)(x) = f(x) − g(x)

(αf)(x) = αf(x)

(fg)(x) = f(x).g(x)

(f / g)(x) = f(x) / g(x)

gof(x) = g(f(x))

ho(gof)(x) = (hog)of(x)

(gof)^-1(x) = f^-1og^-1(x)

Functions Questions with Solutions

Below added are some solved functions questions with solution based on the concept on Functions.

1. Find the inverse of the function f(x) = 4x − 3.

To find the inverse, interchange x and y and solve for y.

x = 4y − 3

Solve for y:

y = (x + 3)/4

So, the inverse function is f^-1(x) = (x + 3)/4

2. Let f(x) = 2x + 3 and g(x) = x². Find (f + g)(x).

Given functions are

f(x) = 2x + 3

g(x) = x²

So, now we need to find (f + g)(x),

(f + g)(x) = f(x) + g(x)

(f + g)(x) = (2x + 3) + x²

So, (f + g)(x) = 2x + 3 + x².

3. If f(x) = sin(x) and g(x) = cos(x), determine (f − g)(x).

Given functions are

f(x) = sin(x)

g(x) = cos(x)

So, we need to need to find (f−g)(x),

(f − g)(x) = f(x) − g(x)

(f − g)(x) = sin(x) − cos(x)

So, (f − g)(x) = sin(x) − cos(x).

4. Given f(x) = 3x², compute (2f)(x).

Given function is

f(x) = 3x²

So, we need to calculate (2f)x

(2f)x = 2.f(x)

= 2.3x²

= 6x²

So, (2f)x = 6x².

5. For f(x) = x + 1 and g(x) = x − 2, what is (f⋅g)(x)?

Given functions are

f(x) = x + 1

g(x) = x − 2

So, we need to calculate (f⋅g)(x),

(f⋅g)(x) = f(x)⋅g(x)

(f⋅g)(x) = (x + 1)(x − 2)

(f⋅g)(x)=x² −2x + x − 2

(f⋅g)(x) = x² − x − 2

So, (f⋅g)(x) = x² − x − 2.

6. If f(x) = 1/x and g(x) = x², find (f/g)(x).

Given functions are

f(x) = 1/x

g(x) = x²

So, we need to calculate (f/g)(x)

(?/?)(?) = ?(?)/?(?)

(f/g)(x) = (1/x)/x²

= 1/x³

So, (f/g)(x) = 1/x³.

7. Let f(x) = 2x + 1 and g(x) = x². Determine g∘f(x).

Given functions are

f(x) = 2x + 1

g(x) = x²

We need to calculate g∘f(x), which involves applying f(x) to g(x).

g∘f(x) = g(f(x))

g∘f(x) = g(2x + 1)

g(x) = x²

Now, substitute 2x+1 for x in the function g(x) = x²:

g∘f(x) = (2x + 1)²

g∘f(x) = 4x² + 4x + 1

So, g∘f(x) = 4x² + 4x + 1.

8. Solve the equation h(x) = 2x² − 5x + 1 for x = 3.

Given equation: h(x) = 2x² − 5x + 1

Substitute x = 3:

h(3) = 2 × 32 − 5 × 3 + 1

=18−15+1

=4

So, h(3) = 4.

9. If f(x) = x² , h(x) = x³ and g(x) = √x, calculate h∘(g∘f)(x).

Given functions are

f(x) = x²

g(x) = √x

h(x) = x³

So, we need to calculate

h∘(g∘f)(x), which involves composing f(x), g(x), and h(x).

First of all, we will find g∘f(x):

g∘f(x)=g(f(x))

g∘f(x)=g(x²)

Now, substitute ?² for x in the function g(x) = √x

g∘f(x) = √x²

g∘f(x) = x

After first, now we will calculate h(x) with g∘f(x):

h∘(g∘f)(x) = h(x)

h∘(g∘f)(x) = x³

So, h∘(g∘f)(x) = x³.

10. If f(x) = 2x + 3 and g(x) = 5x − 2 are inverse functions, what is f-1(x)?

Given function is

g(x) = 5x − 2

Let y = g(x):

y = 5x − 2

Now, solve for x in terms of y:

y + 2 = 5x

x = (y+2)/5

This represents the inverse function g^-1(x).

Therefore, f^-1(x) = g^-1(x) = 5x + 2.

Related Articles

• Types of Functions
• Domain and Range
• Composition of Function

Functions Questions for Practice : Unsolved

Try out the following questions based on the function.

FAQs on Functions Questions with Solutions

List the various type of Functions.

The various type of Function are following:-

• Many-one function
• One-one function
• Into function
• Onto function
• Bijective function

What is the condition for a function to be invertible?

For a function to be invertible, it needs to be bijection function only.

When we can say that two functions are equal?

Two functions can be equal if following conditions hold true:

• Their domain should be same.
• Their co-domain and range should be same.
• For each element in the domain, both the functions should have the same image.

What do you mean by constant functions?

Constant functions are functions which return same output every time irrespective of the input.

What do you mean by domain?

Domain are the set of values which can be taken as input in the function.