1. ## function inverses

1. Find the equation of the inverse for each function. State which of the following inverses are functions.
2. a. f(x) = 1/2 x - 3
3. b. f(x) = 4x 2 - 5
c. A(r) = 4(Pi)r2

2. Originally Posted by euclid2
1. Find the equation of the inverse for each function. State which of the following inverses are functions.
2. a. f(x) = 1/2 x - 3
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.