F(x) Question

Question

For f(x)=³√4x-7

 

A. Name two conditions for existence of f^-1(x)

 

B. Find the inverse of f(x)

Answer 1

Two conditions for the f^-1(x) to exist.

 

1) The function must be in the domain.

 

Since we have a cube-root, negative and positive numbers are allowed to be under the cube root.  So the domain is all real numbers.

 

2) f^-1(x) is a reflection of f(x) over the line y=x.

 

 

To find the inverse, we simply switch places with x and y.

 

if

 

y = ³ √(4x - 7) ,

 

 

then

 

x = ³ √(4y - 7)

 

x = (4y - 7)^1/3

 

 

Now we solve for y.

 

Cube both sides of equation.

 

x³ = 4y - 7

 

 

Add 7 on both sides of equation.

 

x³ + 7 = 4y

 

 

Divide both sides of equation by 4.

 

(x³ + 7) / 4 = y