**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^{-1}2(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.

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