MHB How can inverse functions help us find the range algebraically?

[mathdad]

Find the range algebraically.

y = 125 - 12x

There is a way to do this by finding the inverse function. Can someone show me how to apply the idea of inverse functions to find the range?

 

[skeeter]

$f(x)=125-12x$, a 1-1 function

inverse is $f^{-1}(x) = \dfrac{125-x}{12}$

the domain of $f^{-1}(x)$ is $\mathbb{R} \implies$ range of $f(x)$ is $\mathbb{R}$

 

[mathdad]

This function is one-to-one. Can we find the inverse of functions that are not one-to-one?

You found the inverse of the given function.

You then said that the domain of the inverse is the range of the original function.

Correct?

Can this method be applied to all functions?

 

[skeeter]

Finding the inverse of a function is not always easy.

Sometimes the domain of the function in question must be restricted to force a 1-1 situation.

This is the reason why graphical analysis is so important. More complex graphs require the concepts of calculus to analyze.

 

[mathdad]

You said:

"Sometimes the domain of the function in question must be restricted to force a 1-1 situation."

I am not too clear in terms of restricting a function "to force a 1-1 situation."

What do you mean by this statement which is so common in textbooks?