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How can I find the inverse of f^-1?

 Oct 9, 2017
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\(f(x)=\sqrt{5-x}+3\) 

 

Changing this to its inverse requires a few steps.

 

1. Change \(f(x)\) to \(y\).

 

This step is pretty simple. We are switching from function notation to y=-notation. \(f(x)=\sqrt{5-x}+3\) changes to \(y=\sqrt{5-x}+3\)

 

2. Interchange x and y

 

This step is also quite simple; replace all instances of x with y and all instance of y with x.

 

\(y=\sqrt{5-x}+3\) changes to \(x=\sqrt{5-y}+3\)

 

3. Solve for y

 

This step is the hardest. Transform the equation into the form of y=.

 

\(x=\sqrt{5-y}+3\) Subtract 3 on both sides.
\(x-3=\sqrt{5-y}\) Square both sides to eliminate the square root.
\((x-3)^2=\left(\sqrt{5-y}\right)^2\) Expand the left hand side knowing that \((a-b)^2=a^2-2ab+b^2\).
\(x^2-6x+9=5-y\) Subtract 5 from both sides.
\(-y=x^2-6x+4\) Divide by -1.
\(y=-x^2+6x-4\)  
   

 

4. Consider Whether the Inverse is actually a Function

 

In this case, it is a function, so we are OK.

 Oct 9, 2017

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