EVALUATING COMPOSITE FUNCTIONS

Example 1 :

Given f(t) = t² - t and h(x) = 3x + 2 , evaluate (f ∘ h)(1).

Solution :

(f ∘ h)(1) = f[h(1)] ----(1)

Let us evaluate h(1).

h(x) = 3x + 2

h(1) = 3(1) + 2

h(1) = 5

By applying the value of h(1) = 5 in (1) , we get

(f ∘ h)(1) = f(5)

Finding the value of f(5), we get

f(5) = 5² - 5

f(5) = 25 - 5

f(5) = 20

Example 2 :

Given each pair of functions

f(x) = 4x + 8, g(x) = 7 - x²

calculate

(i)  f (g (0)) and (ii)  g ( f (0))

Solution :

f(x) = 4x + 8, g(x) = 7 - x²

Evaluating  f (g (0)) :

(i)  f (g (0))  -----(1)

g(0) = 7 - 0²

g(0) = 7

Applying g(0) = 7 in (1), we get

f(g(0)) = f(7)

f(7) = 4(7) + 8

f(7) = 28 + 8

f(7) = 36

(ii) Evaluating g (f (0)) :

g ( f (0)) ----(2)

f(0) = 4(0) + 8

f(0) = 8

Applying f(0) = 8 in (2), we get

g(f(0)) = g(8)

 g(8) = 7 - 8²

= 7 - 64

= -57

Example 3 :

Given each pair of functions

f(x) = 5x + 7, g(x) = 4 - 2x²

calculate

(i)  f (g (0)) and (ii)  g (f (0))

Solution :

Evaluating  f (g (0)) :

(i)  f (g (0))  -----(1)

g(0) = 4 - 2(0)²

g(0) = 4

Applying g(0) = 4 in (1), we get

f(g(0)) = f(4)

f(4) = 5(4) + 7

f(4) = 20 + 7

f(4) = 27

(ii) Evaluating g (f (0)) :

g ( f (0)) ----(2)

f(0) = 5(0) + 7

f(0) = 7

Applying f(0) = 7 in (2), we get

g(f(0)) = g(7)

 g(7) = 4 - 2(7²)

= 4 - 2(49)

= 4 - 98

= -94

Example 4 :

Given that f(x) = 3x - 4 and g(x) = 7 - x² calculate.

(i) (f ∘ g)(2)

(ii)  (g ∘ f)(2)

(iii)  (f ∘ g)(-2)

(iv)  (g ∘ f)(-2)

Solution :

(i) (f ∘ g)(2) = f[g(2)]

(ii)  (g ∘ f)(2) = g[f(2)]

= g[3(2) - 4]

= g[6 - 4]

= g[2]

Now in the function g(x), instead of x we will apply 2.

= 7 - 2² 

= 7 - 4

(g ∘ f)(2) = 3

(iii)  (f ∘ g)(-2) = f[g(-2)]

g(x) = 7 - x²

= f [7 - (-2)²]

= f [7 - 4]

= f [3]

Now in the function f(x), we will replace x by 3.

 f(x) = 3x - 4

= 3(3) - 4

= 9 - 4

= 5

(iv)  (g ∘ g)(-2) = g[g(-2)]

g[g(2)] = g[7 - (-2)²] 

= g[7 - 4]

= g[3]

Now in the function g(x), instead of x we will apply 3.

g(3) = 7 - 3²

= 7 - 9

= -2

Example 5 :

Use the graphs below to compute the following values.

Solution :

a)  (g ∘ f) (1) = g[f(1)]

The value of f(1) from the graph of f(x) is 2.

= g[2]

The value of g(2) from the graph of g(x) is 3

= 3

So, the value of (g ∘ f) (1) is 3.

b)  (f ∘ g) (3) = f[g(3)]

The value of g(2) from the graph of g(x) is 0.

= f[0]

The value of f(0) is 4.

So, the value of (f ∘ g) (3) = 4.

c)  (g ∘ f) (2) = g[f(2)]

The value of f(2) is 3.

= g(3)

The value of g(3) is 0.

= 0

(g ∘ f) (2) = 0

d)  (f ∘ g) (0) = f[g(0)]

= f[0]

= 4

e)  (f ∘ f) (1) = f[f(1)]

= f[2]

= 3

f)  (g ∘ g) (1) = g[g(1)]

= g[3]

= 0

Example 6 :

For each of the function pairs, compute (g ∘ f)(0), (f ∘ g)(1/2) and (f ∘ f)(-2)

a)  f(x) = 4 - 3x , g(x) = |x|

b)  f(x) = 4x + 5, g(x) = √x

Solution :

a)  f(x) = 4 - 3x , g(x) = |x|

(f ∘ g)(x) = f [g(x)]

= f [ |x| ]

(f ∘ g) (x) = 4 - 3|x|

Evaluting (f ∘ g)(1/2) :

(f ∘ g)(1/2) = 4 - 3|1/2|

= 4 - 3/2

= (8 - 3)/2

= 5/2

(g ∘ f)(x) = g [f(x)]

= g[4 - 3x]

(g ∘ f)(x) = |4 - 3x|

Evaluating (g ∘ f)(0) :

= |4 - 3(0)|

= |4 - 0|

= 4

(f ∘ f)(x) = f [ f(x) ]

= f [4 - 3x]

(f ∘ f)(x) = 4 - |4 - 3x|

Evauating  (f ∘ f)(-2) :

= 4 - |4 - 3(0)|

= 4 - 4

= 0

b)  f(x) = 4x + 5, g(x) = √x

(f ∘ g)(x) = f [g(x)]

= f[√x]

= 4√x + 5

Evaluting (f ∘ g)(1/2) :

= 4 √(1/2) + 5

= 4/√2 + 5

(g ∘ f)(x) = g [f(x)]

= g(4x+ 5)

= √(4x + 5)

Evaluating (g ∘ f)(0) :

= √(4(0) + 5)

= √5

(f ∘ f)(x) = f [ f(x) ]

= f[4x + 5]

= 4(4x + 5) + 5

= 16x + 20 + 5

= 16x + 25

Evauating  (f ∘ f)(-2) :

= 16(0) + 25

= 25