Question 13: Determine the inverse function of each of the following functions.
a. y = f(x) = x + 5
b. y = f(x) = 2x + 7
c. y = f(x) = 2(x – 4)
d. y = f(x) = (x+4)/2

Solution

The inverse of a function ‘f’ is a function that maps the range of ‘f’ to the domain of ‘f’.
So, if {(a,1), (b,2), (c,3)} is a function, its inverse is {(1,a), (2,b), (3,c)}.
Therefore, when the function f is applied to an input x and it gives a result y, then the inverse function g will be such that when it is applied to y, will give the result x again.

a. y = f(x) = x + 5
Now the function f(x) = x + 5 takes an output x + 5 in response to input x. So, the inverse function must take output x in response to input x + 5.
f-1(x+5) = x
Put x + 5 = z ⇒ x = z – 5, therefore,
f-1(z) = z – 5.
Substitute x as input instead of z to obtain the required inverse function.
f-1(x) = x – 5

b. y = f(x) = 2x + 7
Now the function f(x) = 2x + 7 takes an output 2x + 7 in response to input x. So, the inverse function must take output x in response to input 2x + 7.
f-1(2x+7) = x
Put 2x + 7 = z ⇒ x = (z – 7)/2, therefore,
f-1(z) = (z – 7)/2.
Substitute x as input instead of z to obtain the required inverse function.
f-1(x) = (x – 7)/2

c. y = f(x) = 2(x – 4)
Now the function f(x) = 2(x – 4) takes an output 2(x – 4) in response to input x. So, the inverse function must take output x in response to input 2(x – 4).
f-12(x – 4) = x
Put 2(x – 4) = z ⇒ x – 4 = z/2 or x = z/2 + 4, therefore,
f-1(z) = z/2 + 4.
Substitute x as input instead of z to obtain the required inverse function.
f-1(x) = x/2 + 4

d. y = f(x) = (x+4)/2
Now the function f(x) = (x + 4)/2 takes an output (x + 4)/2 in response to input x. So, the inverse function must take output x in response to input (x+4)/2.
f-1(x + 4)/2 = x
Put (x + 4)/2 = z ⇒ (x + 4) = 2z Or x = 2z – 4, therefore,
f-1(z) = 2z – 4.
Substitute x as input instead of z to obtain the required inverse function.
f-1(x) = 2x – 4.