1. ## Inverse Equation

How would I find out the inverse for these equations?

F(x) = 2√(x+3)
G(x) = 2^(x+1)+4
Y= 2(x-5)^3

2. Hello, takkun0486!

Do you know the procedure for finding an inverse function?

[1] Replace the function with y.
[2] "Switch" the x's and y's.
[3] Solve for y.

How would I find out the inverses for these functions?

$(a)\;\;F(x) \:= \:2\sqrt{x+3}$
$(b)\;\;G(x) \:= \:2^{x+1} + 4$
$(c)\;\;Y\:= \:2(x-5)^3$
$(a)$ We have: $[1]\;\;y\;=\;2\sqrt{x+3}$

"Switch": $[2]\;\;x\;=\;2\sqrt{y + 3}$

Solve for y: . $[3]\;\;\frac{x}{2}\;=\;\sqrt{y + 3}$

Square: . $\left(\frac{x}{2}\right)^2\;=\;(\sqrt{y + 3})^2\quad\Rightarrow\quad \frac{x^2}{4}\;=\;y + 3$

Then: . $\frac{x^2}{4} - 3 \;= \;y$

Therefore: . $f^{-1}(x)\;=\;\frac{x^2}{4} - 3$

$(b)$ We have: . $[1]\;y \:= \:2^{x+1} + 4$

Switch: . $[2]\;\;x\;=\;2^{y+1} + 4$

Solve for y: . $[3]\;\;x - 4\:=\:2^{y+1}$

Take logs: . $\ln(x - 4)\:=\:\ln(2^{y+1})\quad\Rightarrow\quad \ln(x - 4)\:=\y + 1)\ln(2) " alt="\ln(x - 4)\:=\:\ln(2^{y+1})\quad\Rightarrow\quad \ln(x - 4)\:=\y + 1)\ln(2) " />

Then: . $\frac{\ln(x - 4)}{\ln(2)}\:=\:y + 1\quad\Rightarrow\quad\frac{\ln(x-4)}{\ln(2)} - 1\:=\:y$

Therefore: . $G^{-1}(x)\;=\;\frac{\ln(x-4)}{\ln(2)} - 1$

$(c)$ We have: . $y \:=\:2(x - 5)^3$

Switch: . $x \;= \;2(y - 5)^3$

Solve for y: . $\frac{x}{2}\;=\;(y - 5)^3$

Take cube roots: . $\sqrt[3]{\frac{x}{2}}\;=\;y - 5$

Then: . $\sqrt[3]{\frac{x}{2}} + 5 \;= \;y$

Therefore: . $Y^{-1}\;=\;\sqrt[3]{\frac{x}{2}} + 5$ . . . or: $Y^{-1} \;= \;\frac{\sqrt[3]{4x}}{2} + 5$