Functions Practice Questions : Solved and Unsolved

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.

Below are some solved function questions with solutions based on the concept of functions.

Question 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

Question 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.

Question 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).

Question 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.

Question 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.

Question 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.

Question 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.

Question 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.

Question 9. If f(x) = x, h(x) = x,³ 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.

Question 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.

Practice Questions Unsolved

Try out the following questions based on the function.

Q1. Let f(x) = x + 4 and g(x) = 3x − 1. Find (f + g)(x).

Q2. If f(x) = cos⁡(x) and g(x) = sin⁡(x), determine (f - g)(x).

Q3. Given f(x) = 2x³ - x, compute (-3f)(x).

Q4. For f(x) = x − 2 and g(x) = 1/x​, what is (f ⋅ g)(x)?