FINDING COMPOSITION OF FUNCTIONS

Example 1 :

If f(x) = 3x - 5 and g(x) = x², find (f∘g) (x).

Solution :

(f∘g) (x) = f ( g(x) )

Here the value of g(x) is x².

(f∘g) (x) = f ( x² )

f (x²) looks like f(x), here x = x²

So, in the function f(x) put x², where we see x.

(f∘g) (x) = 3x² - 5

Example 2 :

Let f(x) = -3x + 7 and g(x) = 2x² - 8, find (f∘g) (x) and (g∘f) (x).

Solution :

(i)  f(x) = -3x + 7 and g(x) = 2x² - 8

(f∘g) (x) = f (g(x) )

= f(2x² - 8)

Now in f(x), x can be replaced by 2x² - 8.

f(2x² - 8) = -3(2x² - 8) + 7

Distributing -3 and combining like terms, we get

= -6x² + 24 + 7

= -6x² + 31

(ii)  (g∘f) (x) = g( f(x) )

= g(-3x + 7)

Now in g(x), x can be replaced by -3x + 7

g(-3x + 7) = 2(-3x + 7)² - 8

Using the algebraic identity, expanding it.

g(-3x + 7) = 2(9x² - 42x + 49) - 8

= 18x² - 84x + 98 - 8

= 18x² - 84x + 90

Example 3 :

If f(x) = -9x - 9 and g(x) = √(x - 9), find (f∘g) (x).

Solution :

(f∘g) (x) = f( g(x) )

= f( g(x) )

Example 4 :

If f(x) = -2x + 1 and g(x) = √(x² - 9), find

(i)  (f∘g) (x) and 

(ii)  (g∘f) (x)

Solution :

Finding (f∘g) (x) :

Finding (g∘f) (x) :

Example 5 :

If f(x) = -2x + 1 and g(x) = √(x² - 5), find

(i)  (f∘g) (x) and 

(ii)  (g∘f) (x)

Solution :

(i)  Finding (f∘g) (x) :

(f∘g) (x) = f ( g(x) )

Here the value of g(x) is √(x² - 5).

= f(√(x² - 5))

In the function f(x), we will apply x as √(x² - 5).

f(g(x)) = -2(x² - 5) + 1

= -2x² + 10 + 1

= -2x² + 11

(ii)  Finding (g∘f) (x) :

(g∘f) (x) = g ( f(x) )

Here the value of f(x) is -2x + 1.

g ( f(x) ) = g(-2x + 1)

Here g(-2x + 1) exactly matches with g(x). So, apply x as -2x + 1 in the function g(x).

=  √((-2x + 1)² - 5)

=  √(4x² - 4x + 1 - 5)

=  √(4x² - 4x - 4)

=  √4(x² - x - 1)

=  2√(x² - x - 1)

Example 6 :

A banquet hall charges $975 to rent a reception room, plus $39.95 per person. Next month, however, the banquet hall will be offering a 20% discount off the total bill. Express this discounted cost as a function of the number of people attending.

Solution :

Let x be the number of persons.

Charge for reception room = $975

Cost per person = $39.95

Total charge = 975 + 39.95x

Discount = 20%

After reducing 20% 

= 80% of (975 + 39.95x)

= 0.80(975 + 39.95x)

= 780 + 31.96x

Example 7 :

For a car travelling at a constant speed of 80 km/hr, the distance driven d kilometers is represented by

d(t) = 80t

where t is is the time in hours. The cost of gasoline in dollars, for the drive is represented by 

C(d) = 0.09d

a) determine C(d(5)) numerically and interpret your result.

b)  Describe the relationship represented by C(d(t))

Solution :

d(t) = 80t

C(d) = 0.09d

a) C(d(5)) 

Evaluating d(5), we get

d(5) = 80(5)

= 400

C(400) = 0.09(400)

= 36

b) 

C(d(t)) = C(80t)

= 0.09(80t)

= 7.2t

C(d(t)) represents the relationship between the time driven and the cost of gasoline.

Example 8 :

The function p(d) = 0.03d + 1 approximates the pressure (in atmospheres) at a depth of d feet below sea level. The function d(t) = 60t represents the depth (in feet) of a diver t minutes after beginning a descent from sea level, where 0 ≤ t ≤ 2.

a. Find p(d(t)). Interpret the terms and coefficient.

b. Evaluate p(d(1.5)) and explain what it represents.

Solution :

p(d) = 0.03d + 1

d(t) = 60t

a)  p(d(t)) = p(60t)

Applying 60t in the function p(d), we get

= 0.03(60t) + 1

= 1.8t + 1

b) 

p(d(1.5)) = 1.8(1.5) + 1

= 2.7 + 1

= 3.7