#1**+1 **

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

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.

TheXSquaredFactor
Oct 9, 2017