Question

Find f^{-1}(x) for the function below. Do not put spaces between your characters. If no inverse exists then type "none".f(x)=\frac{2x+3}{5x+4}The numerator of f^{-1}(x)=AnswerThe denominator of f^{-1}(x)=Answer

Answer 1

We want to find the inverse of the following function

$f(x)=\frac{2x+3}{5x+4}$

To find the expression of f^-1(x) we begin by changing f(x) for y. So we get

$y=\frac{2x+3}{5x+4}$

Now, we interchange x and y variables. So we get

$x=\frac{2y+3}{5y+4}$

The goal is to solve this equation for variable y. We begin by multiplying both sides by 5y+4. So we get

$x\cdot(5y+4)=2y+3$

Now we subtract 2y on both sides, so we get

$x\cdot(5y+4)\text{ -2y=3}$

Now we distribute the product on the left, so we get

$5\cdot xy+4\cdot x\text{ -2y=3}$

Now we group the terms that have the y variable and factor it out. So we get

$y(5x\text{ -2)+4x=3}$

now, we subtract 4x on both sides, so we get

$y(5x\text{ -2)=3 -4x}$

Finally, we divide both sides by 5x-2. We get

$y=\frac{3\text{ -4x}}{5x\text{ -2}}$

Now, we replace y with f^-1(x). So we get

$f^{\text{ -1}}(x)=\frac{3\text{ -4x}}{5x\text{ -2}}$

this means that the numerator of f inverse is 3 -4x and the denominator of f inverse is 5x-2