FINDING INVERSE OF A FUNCTION

Practice Questions

Question 1 :

Suppose f(x) = 2x + 3.

(a) Evaluate f⁻¹(11).

(b) Find a formula for f⁻¹(y).

Solution :

In order to find the inverse function, first let us prove the given function is one to one.

The domain of the function f(x) is all real values, each real values of x is associated with unique elements of y. So it is one to one function.

 y  =  f(x)  =  2x + 3

 2x + 3  =  y

2x  =  y - 3

x  =  (y - 3)/2

Replace x by f^-1 (y) and y by x.

f^-1(y)  =  (y - 3)/2

Question 2 :

Suppose the domain of f is the interval [0, 2], with f defined on this domain by the equation f(x) = x². 

(a) What is the range of f ?

(b) Find a formula for the inverse function f ⁻¹.

(c) What is the domain of the inverse function f ⁻¹?

(d) What is the range of the inverse function f ⁻¹?

Solution :

f(x) = x²

From this, we know that range is [0, 4].

(b) Find a formula for the inverse function f ⁻¹.

From the table given above, we know that each values is associated with unique elements. So it is one to one.

y = x²

x  =  √y

Here x can be replaced by f^-1(y), so f^-1(y)  =  √y

(c) What is the domain of the inverse function f ⁻¹?

Domain of f ⁻¹  =  Range of f  =  [0, 4]

(d) What is the range of the inverse function f ⁻¹?

Range of f ⁻¹  =  Domain of f  =  [0, 2]

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