To find the inverse of
f ( x ) , f(x), f ( x ) , we first need to replace
f ( x ) f(x) f ( x ) with
y . y. y . f ( x ) = 4 − 2 x → y = 4 − 2 x \begin{gathered}
f(x)=4-2\sqrt{x} \quad \rightarrow \quad y=4-2\sqrt{x}
\end{gathered} f ( x ) = 4 − 2 x → y = 4 − 2 x
Next step is to switch
x x x and
y y y in the function rule.
y = 4 − 2 x ⟶ switch x = 4 − 2 y
y=4-2\sqrt{x} \quad \stackrel{\text{switch}}{\longrightarrow} \quad x=4-2\sqrt{y}
y = 4 − 2 x ⟶ switch x = 4 − 2 y
Now we need to solve for
y . y. y . The resulting equation will be the inverse of the given function.
x = 4 − 2 y x=4-2\sqrt{y} x = 4 − 2 y x − 4 = - 2 y x-4=\text{-} 2\sqrt{y} x − 4 = - 2 y x − 4 - 2 = y \dfrac{x-4}{\text{-} 2}=\sqrt{y} - 2 x − 4 = y ( x − 4 - 2 ) 2 = ( y ) 2 \left(\dfrac{x-4}{\text{-} 2}\right)^2=\left(\sqrt{y}\right)^2 ( - 2 x − 4 ) 2 = ( y ) 2 ( x − 4 ) 2 ( - 2 ) 2 = ( y ) 2 \dfrac{(x-4)^2}{(\text{-} 2)^2}=\left(\sqrt{y}\right)^2 ( - 2 ) 2 ( x − 4 ) 2 = ( y ) 2 ( x − 4 ) 2 4 = ( y ) 2 \dfrac{(x-4)^2}{4}=\left(\sqrt{y}\right)^2 4 ( x − 4 ) 2 = ( y ) 2 1 4 ( x − 4 ) 2 = ( y ) 2 \dfrac{1}{4}(x-4)^2=\left(\sqrt{y}\right)^2 4 1 ( x − 4 ) 2 = ( y ) 2 1 4 ( x − 4 ) 2 = y \dfrac{1}{4}(x-4)^2=y 4 1 ( x − 4 ) 2 = y
y = 1 4 ( x − 4 ) 2 y=\dfrac{1}{4}(x-4)^2 y = 4 1 ( x − 4 ) 2
Finally, to indicate that this is the inverse of
f ( x ) , f(x), f ( x ) , we will replace
y y y with
f - 1 ( x ) . f^{\text{-} 1}(x). f - 1 ( x ) . y = 1 4 ( x − 4 ) 2 → f - 1 ( x ) = 1 4 ( x − 4 ) 2 \begin{gathered}
y=\dfrac{1}{4}(x-4)^2 \quad \rightarrow \quad {\color{#0000FF}{f^{\text{-} 1}(x)}}=\dfrac{1}{4}(x-4)^2
\end{gathered} y = 4 1 ( x − 4 ) 2 → f - 1 ( x ) = 4 1 ( x − 4 ) 2 