Question

How do you find the inverse of `h(X)= 5 / (2x + 3)` and is it a function?

Answer 1

The inverse of a function can be found algebraically by switching the x and y values

Explanation:

`y = 5/(2x + 3)`

`x = 5/(2y + 3)`

`x(2y + 3) = 5`

`2y + 3 = 5/x`

`2y = (5 - 3x)/x`

`y = (5 - 3x)/(2x)`

`h^-1(x) = (5 - 3x)/(2x)`

Here are a few things to remember when finding the inverse of a function:

1. The y must be isolated (all alone on one side of the equation).

2. Don't forget the #h^-1(x) notation. I have been docked marks before from forgetting to include this element in my answer.

3. The inverse of a function can be found graphically by reflecting the original function over the line `y = x`

The first graph below is of the original function. The second is of the inverse.

graph{y = 5/(2x + 3) [-10, 10, -5, 5]}

graph{y = (5 - 3x)/(2x) [-4.933, 4.933, -2.466, 2.467]}

Practice exercises:

1. Indicate the inverses of the following functions, and then state whether or not they are functions.

a) `ƒ(x) = 2x + 7`

b) `h(x) = 3x^2 - 5x + 1`

c) `j(x) = sqrt(2x - 4) + 5`

d) `k(x) = (3x - 2)/(5x + 4)`

`2.` The following graph is of function `ƒ^(-1)(x)`. State the equation of `ƒ(x)`

graph{y = 1/3x + 2 [-9.51, 9.51, -4.755, 4.755]}

Good luck!