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

Mar 20, 2016

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

#### Explanation:

$y = \frac{5}{2 x + 3}$

$x = \frac{5}{2 y + 3}$

$x \left(2 y + 3\right) = 5$

$2 y + 3 = \frac{5}{x}$

$2 y = \frac{5 - 3 x}{x}$

$y = \frac{5 - 3 x}{2 x}$

${h}^{-} 1 \left(x\right) = \frac{5 - 3 x}{2 x}$

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) $h \left(x\right) = 3 {x}^{2} - 5 x + 1$

c) $j \left(x\right) = \sqrt{2 x - 4} + 5$

d) $k \left(x\right) = \frac{3 x - 2}{5 x + 4}$

$2.$ 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!