FIND COMPOSITION OF TWO FUNCTIONS

Problem 1 :

Given f(x) = -9x + 3 and g(x) = x⁴, find (f ∘ g)(x)

Solution :

(f ∘ g)(x) = f(g(x))

= f(x⁴)

= -9(x⁴) + 3

(f ∘ g)(x) = -9x⁴ + 3

Problem 2 :

Given f(x) = 2x – 5 and g(x) = x + 2, find (f ∘ g)(x)

Solution :

(f ∘ g)(x) = f(g(x))

= f(x + 2)

= 2(x + 2) – 5

= 2x + 4 – 5

= 2x - 1

 (f ∘ g)(x) = 2x - 1

Problem 3 :

Given f(x) = x² + 7 and g(x) = x - 3, find (f ∘ g)(x)

Solution :

(f ∘ g)(x) = f(g(x))

= f(x - 3)

= (x - 3)² + 7

= x² + (3)² – 2(x)(3) + 7

= x² + 9 - 6x + 7

 (f ∘ g)(x) = x² - 6x + 16

Problem 4 :

Given f(x) = 4x + 3 and g(x) = x², find (g ∘ f)(x)

Solution :

(g ∘ f)(x) = g(f(x))

= g(4x + 3)

= (4x + 3)²

= (4x)² + (3)² + 2(4x)(3)

= 16x² + 9 + 24x

 (g ∘ f)(x) = 16x² + 24x + 9

Problem 5 :

Given f(x) = x – 1 and g(x) = x² + 2x - 8, find (g ∘ f)(x)

Solution :

(g ∘ f)(x) = g(f(x))

= g(x - 1)

= (x – 1)² + 2(x – 1) - 8

= (x)² + (1)² – 2(x)(1) + 2x – 2 - 8

= x² + 1 - 2x + 2x - 10

 (g ∘ f)(x) = x² - 9

Problem 6 :

Use f(x) = 2x + 3 adn g(x) = 1 - x² to evaluate the following expressions.

a)  f (g(0))        b)  g (f(4))        c)  (f ∘ g) (-8)

d)  (g ∘ g) (1/2)     e)  (f ∘ f^-1) (1)      f)  (g ∘ g) (2)

Solution :

f(x) = 2x + 3 adn g(x) = 1 - x²

a)  f (g(0))

g(0) = 1 - 0²

= 1

Applying the value of g(0), we get

f (g(0)) =  f (1)

= 2(1) + 3

= 2 + 3

= 5

b)  g (f(4))

f(4) = 2(4) + 3

= 8 + 3

= 11

Applying the value of f(4), we get

g (f(4)) = g(11)

= 1 - 11²

= 1 - 121

= -120

c)  (f ∘ g) (-8) = f [g(8)]

g(8) = 1 - 8²

= 1 - 64

= -63

Applying the value of g(8), we get

= f(-63)

= 2(-63) + 3

= -126 + 3

= -123

d)  (g ∘ g) (1/2) = g[g(1/2)]

g(1/2) = 1 - (1/2)²

= 1 - (1/4)

g(1/2) = 3/4

Applying the value of g(1/2), we get

g(3/4) = 1 - (3/4)²

= 1 - 9/16

= (16-9)/16

= 7/16

e)  (f ∘ f^-1) (1) = f[f^-1) (1)]

Evaluating f^-1 (x) :

Let y = 2x + 3

y - 3 = 2x

x = (1/2) (y - 3)

Replace x by f^-1 (x) and y by x.

f^-1 (x) = (1/2) (x - 3)

f^-1 (1) = (1/2) (1 - 3)

= -2/2

= -1

By applying the value of f^-1 (1) = -1, we get

f[f^-1) (1)] = f(-1)

= (1/2)(-1-3)

= -4/2

= -2

f)  (g ∘ g) (2) = g[g(2)]

g(2) = 1 - 2²

= 1 - 4

= -3

Applying the value of g(2), we get

= g(-3)

g(-3) = 1 - (-3)²

= 1 - 9

= -8

Problem 7 :

Given f = {(0, 1) (1, 2) (2, 5) (3, 10)} and g = {(2, 0)(3, 1)(4, 2)(5, 3)(6, 4)} determine the follwoing

a) (g ∘ f) (2) = g[f(2)]

When input is 2, the output will be 5.

= g(5)

When the input is 5, the output will be 3.

b) (f ∘ f) (1) = f[f(1)]

When the input is 1, the output is 2.

= f(2)

= 0

c)  (f ∘ g) (5) = f[g(5)]

= f[3]

= 10

d)  (f ∘ g) (0) = f(g(0))

There is no input 0 for the function g. So, it is undefined.

e)  (f ∘ f^-1) (2) = f[f^-1(2)]

f^-1(2) = 1

= f(1)

= 2

f)  (g^-1 ∘ f) (1) = g^-1 (f(1))

= g^-1 (2)

= 4

Problem 8 :

Use the graphs of f and g to evaluate each expression.

a)  f(g(2))

b)  (g ∘ g) (-2)

c)  (g ∘ f) (4)

d)  (f ∘ f) (2)

a)  f(g(2)) = f(5)

= 5

b)  (g ∘ g) (-2) = g(g(-2))

= g(1)

= 4

c)  (g ∘ f) (4) = g(f(4))

= g(3)

= 5

d)  (f ∘ f) (2) = f(f(2))

= f(-1)

= undefined.