+2446 \(f(x)=\sqrt{5-x}+3\)
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.
