SOLVING WORD PROBLEMS ON COMPOSITION OF FUNCTIONS

Problem 1 :

If f(x) = 2x² + 5 and g(x) = 3x + a, find a so that the graph of f ∘ g crosses the y - axis at 23.

Solution :

Given, f(x) = 2x² + 5 and g(x) = 3x + a

To find a :

f ∘ g = f[g(x)]

= 2x² + 5

= 2(3x + a)² + 5

f ∘ g crosses the y - axis at 23. So, x axis is 0.

 2(3(0) + a)² + 5 = 23

2a² + 5 = 23

2a² = 23 - 5

2a² = 18

a² = 18/2

a² = 9

a = ± 3

Problem 2 :

If f(x) = 3x² - 7 and g(x) = 2x + a, find a so that the graph of f ∘ g crosses the y - axis at 68.

Solution :

Given, f(x) = 3x² - 7 and g(x) = 2x + a

To find a :

f ∘ g = f[g(x)]

= 3x² - 7

= 3(2x + a)² - 7

f ∘ g crosses the y - axis at 68. So, x axis is 0.

 3(2(0) + a)² - 7 = 68

3a² - 7 = 68

3a² = 68 + 7

3a² = 75

a² = 75/3

a² = 25

a = ± 5

Problem 3 :

The number of bacteria in a refrigerated food product is given by

N(T) = 27T² - 97T  + 51, 3 < T < 33

where T is the temperature of the food.

When the food is removed from the refrigerator, the temperature is given by

T(t) = 4t + 1.7

where t is the time in hours.

a) Find the composite function N(T(t)) :

N(T(t)) = _____

b) Find the time when the bacteria count reaches 142.

Time Needed = _____

Solution :

N(T) = 27T² - 97T  + 51

 Temperature is given by T(t) = 4t + 1.7

N(T) = 27(4t + 1.7)² - 97(4t + 1.7)  + 51

= 27[(4t)² + (1.7)² + 2(4t)(1.7)] - 388t - 164.9 + 51

= 27(16t² + 2.89 + 13.6t) - 388t - 164.9 + 51 

= 432t² + 78.03 + 367.2t - 388t - 164.9 + 51 

N(T) = 432t² - 20.8t - 35.87

432t² - 20.8t - 35.87 = 142

432t² - 20.8t - 35.87 - 142 = 0

432t² - 20.8t - 177.87 = 0

So, the required time = 0.6180.

Problem 4 :

Recall Dolbear's function that defines temperature, F, in Fahrenheit degrees, as a function of the number of chirps per minute, N, is

F = D(N) = 40  + 1/4 N.

a) Solve the equation F = 40 + (1/4) N for N in terms of F.

b) Say that N = g(F) is the function you just found in

c). What is the meaning of this function ? What does it take as inputs and what does it produce as outputs ?

d) How many chirps per minute do we expect when the outside temperature is 82 degrees F?

How can we express this in the notation of the function g ?

Solution :

a.

F = 40  + (1/4) N

N = (F - 40)4 

N = 4F - 160

b. Meaning of this function :

g(F) = 4F - 160

F - Fahrenheit degrees, g(F) = per minute

c.

N = 4F - 160

N = 4(82) - 160

N = 328 - 160

N = 168