- Find the equation of the inverse for each function. State which of the following inverses are functions.
- a. f(x) = 1/2 x - 3
- b. f(x) = 4x 2 - 5
c. A(r) = 4(Pi)r2
a) First, write the equation as
$\displaystyle y=\frac{x}{2}-3$
Interchange the x and y.
$\displaystyle x=\frac{y}{2}-3$
Solve for y.
$\displaystyle 2x=y-6$
$\displaystyle y=2x+6$
$\displaystyle f^{-1}(x)=2x+6$
The inverse is also a linear function.
b) Follow same procedure as (a) above.
$\displaystyle f(x)=y=4x^2-5$
$\displaystyle x=4y^2-5$
$\displaystyle 4y^2=x+5$
$\displaystyle y^2=\frac{x+5}{4}$
$\displaystyle y=f^{-1}(x)=\pm\frac{\sqrt{x+5}}{2}$
This inverse is not a function because any acceptable value of domain x yields two values for range y.
c) Follow same procedure as (a) above.
$\displaystyle A(r)=y=4\pi r^2$
$\displaystyle r=4\pi y^2$
$\displaystyle y^2=\frac{r}{4\pi}$
$\displaystyle y=A^{-1}(r)=\pm\frac{1}{2}\sqrt{\frac{r}{\pi}}$
This inverse is not a fraction because any acceptable value of domain r yields 2 values for range y.