Given the function:
[tex]f\mleft(x\mright)=\sqrt{3x+9}[/tex]
You can find the Inverse Function by following the steps shown below:
1. Rewrite the function using:
[tex]f(x)=y[/tex]
Then:
[tex]y=\sqrt{3x+9}[/tex]
2. Solve for "x":
- Square both sides of the equation, in order to undo the effect of the square root on the right side:
[tex]\begin{gathered} (y)^2=(\sqrt[]{3x+9})^2 \\ y^2=3x+9 \end{gathered}[/tex]
- Apply the Subtraction Property of Equality by subtraction 9 from both sides of the equation:
[tex]\begin{gathered} y^2-(9)=3x+9-(9) \\ \\ y^2-9=3x \end{gathered}[/tex]
- Apply the Division Property of Equality by dividing both sides of the equation by 3:
[tex]\begin{gathered} \frac{y^2-9}{3}=\frac{3x}{3} \\ \\ \frac{y^2-9}{3}=x \\ \\ x=\frac{y^2-9}{3} \end{gathered}[/tex]
3. Swap the variables:
[tex]y=\frac{x^2-9}{3}[/tex]
4. Replace the variable "y" with:
[tex]y=f^{-1}(x)[/tex]
Then, you get:
[tex]f^{-1}(x)=\frac{x^2-9}{3}[/tex]
Hence, the answer is:
[tex]f^{-1}(x)=\frac{x^2-9}{3}[/tex]
