RELATIONS AND FUNCTIONS PRACTICE TEST

To find questions from 1 to 4, please visit the page "Practice Worksheet Relations and Functions with Answers"

Question 5 :

Find the domain of the function f(x) =

Solution :

Let t = √(1 - √(1 - x²))

f(x)  =  √(1 - t)

1 - t ≥ 0

t ≤ 1

By applying the value of t, we get

√(1 - √(1 - x²)) ≤ 1

1 - √(1 - x²)) ≤ 1

Subtracting 1 through out the equation,

- √(1 - x²)) ≤ 0

√(1 - x²)) ≥0

Taking squares on both sides,

(1 - x²)  ≥ 0

 - x² ≥ -1

x ≤ 1

Hence the domain is x ∈ [-1, 1]

Question 6 :

If f (x) = x² , g(x) = 3x and h(x) = x −2 , Prove that (f ∘ g) ∘ h = f ∘ (g ∘ h) .

Solution :

L.H.S :

R.H.S :

Hence proved.

Question 7 :

Let A = {1, 2} and B = {1, 2, 3, 4} , C = {5, 6} and D = {5, 6 ,7, 8} . Verify whether A × C is a subset of B × D?

Solution :

A = {1, 2} and C = {5, 6}

A x C  =  {(1, 5) (1, 6) (2, 5) (2, 6)}  ----(1)

B = {1, 2, 3, 4} and D = {5, 6 ,7, 8}

B x D  =  { (1, 5) (1, 6) (1, 7) (1, 8) (2, 5) (2, 6) (2, 7) (2, 8) (3, 5) (3, 6) (3, 7) (3, 8) (4, 5) (4, 6) (4, 7) (4, 8) }

Hence A x C is the subset of B x D.

Question 8 :

If f(x)  =  (x - 1)/(x + 1), x ≠ 1 show that f(f(x))  =  -1/x, provided x ≠ 0

Solution :

f(x)  =  (x - 1)/(x + 1) (Given)

f(f(x))  =  

Hence proved.

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