Inverse Functions

Find the inverse of f($x$) = $x+4$.

• 1Write $y=x+4$
• 2Swap $x$ and $y$: $x=y+4$
• 3Rearrange for $y$: $y=x-4$

Answer: $f^{-1}$($x$) = $x-4$

Find the inverse of f($x$) = $3x-2$.

• 1Write $y=3x-2$
• 2Swap $x$ and $y$: $x=3y-2$
• 3Rearrange for $y$: add $2$ to both sides: $x+2=3y$. Divide by $3$: $y$ = ($x+2$) ÷ $3$

Answer: $f^{-1}$($x$) = ($x+2$) ÷ $3$

Check that $f^{-1}$($x$) = ($x+2$) ÷ $3$ correctly undoes f($x$) = $3x-2$, using $x=5$.

• 1Find f($5$): $3$($5$) − $2=13$
• 2Feed $13$ into $f^{-1}$: ($13+2$) ÷ $3=\frac{15}{3}=5$
• 3We got back to $5$, the original input.

Answer: Confirmed — $f^{-1}$ undoes f correctly.