determine, find, and graph..

Question

Given f(x)=(x+5)^3. (a) Determine whether f(x) is one-to-one. (b) Find a formula for its inverse. (c) Graph f(x) and f^(-1) (x) on the same axis.

Answer 1

You can graph the function and if it passes the vertical line test, it's one-to-one.

 

 

Does f(a) = f(b) imply that a=b?

 

f(a) = (a+5)^3

f(b) = (b+5)^3

 

If (a+5)^3 = (b+5)^3...

 

Cube root of both sides:

 

a+5 = b+5

 

Subtract 5 from both sides:

 

a = b.   It's one-to-one.

 

Note that the parent function is y = x^3, which is one-to-one.

 

It is shifted 5 units to the left, so it crosses the x axis at (-5, 0).  Down to the left and up to the right.

 

 

To get the inverse function, exchange x and y and solve for y:

 

x = (y+5)^3

 

Take the cube root of both sides:

 

x^(1/3) = y+5

 

y = x^1/3 - 5

 

 

If they are inverses, once you graph them, they should be reflections of each other across the line y=x.