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)=(2x+3)/(5x+4)`

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

`y=(2x+3)/(5x+4)`

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

`x=(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