COMPOSITE FUNCTION WORKSHEET

Problem 1 :

Let f(x) = 3x + 2 and g(x) = 5x. Find (f ∘ g) and (g ∘ f).

Problem 2 :

Let f(x) = 2x² - 3 and g(x) = 4x. Find (f ∘ g) and (g ∘ f).

Problem 3 :

Let f(x) = x + 3 and g(x) = x² - 9. Find (f ∘ g) and (g ∘ f).

Problem 4 :

Let f(x) = 2x - 1, g(x) = 3x and h(x) = x² + 1.

Compute the following :

(i) f ∘ g(-3)

(ii) f ∘ h(7)

(iii) (g ∘ h)(24)

(iv) f{g[h(2)]}

Problem 5 :

Let f(x) = log₁₀x and g(x) = 10^x. Find (f ∘ g) and (g ∘ f).

1. Solution :

f(x) = 3x + 2 and g(x) = 5x

f ∘ g :

f ∘ g = f[g(x)]

= f(5x)

= 3(5x) + 2

= 15x + 2

g ∘ f :

g ∘ f = g[f(x)]

= g(3x + 2)

= 5(3x + 2)

= 15x + 10

2. Solution :

f(x) = 2x² - 3 and g(x) = 4x

f ∘ g :

f ∘ g = f[g(x)]

= f(4x)

= 2(4x)² - 3

= 2(4²x²) - 3

= 2(16x²) - 3

= 32x² - 3

g ∘ f :

g ∘ f = g[f(x)]

= g(2x² - 3)

= 4(2x² - 3)

= 8x² - 12

3. Solution :

f(x) = x + 3 and g(x) = x² - 9

f ∘ g :

f ∘ g = f[g(x)]

= f(x² - 9)

= (x² - 9) + 3

= x² - 9 + 3

= x² - 6

g ∘ f :

g ∘ f = g[f(x)]

= g(x + 3)

= (x + 3)² - 9

= (x + 3)(x + 3) - 9

= x² + 3x + 3x + 9 - 9

= x² + 6x

4. Solution :

f(x) = 2x - 1, g(x) = 3x and h(x) = x² + 1

(i) f ∘ g(-3) :

f ∘ g(-3) = f[g(-3)]

= f[3(-3)]

= f(-9)

= 2(-9) - 1

= -18 - 1

= -19

(ii) f ∘ h(7) :

f ∘ h(7) = f[h(7)]

= f(7² + 1)

= f(49 + 1)

= f(50)

= 2(50) - 1

= 100 - 1

= 99

(iii) g ∘ h(24) :

g ∘ h(24) = g[h(24)]

= g(24² + 1)

= g(576 + 1)

= g(577)

= 3(577)

= 1731

(iv) f{g[h(2)]} :

f{g[h(2)]} = f{g[2² + 1]}

= f{g[4 + 1]}

= f{g(5)}

= f{3(5)}

= f(15)

= 2(15) - 1

= 30 - 1

= 29

5. Solution :

f(x) = log₁₀x and g(x) = 10^x

f ∘ g :

f ∘ g = f[g(x)]

= f(10^x)

= log₁₀10^x

= xlog₁₀10

= x(1)

= x

g ∘ f :

g ∘ f = g[f(x)]

= g(log₁₀x)

= 10^log₁₀^x

= x

Here f ∘ g = g ∘ f = x.

So f(x) and g(x) are inverse to each other. 

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.