Question

Find the inverse of the function below. When typing your answer use the "^" key (shift+6) to indicate an exponent. For example, if we have x squared (x times x) we would type x^2. f(x)= \frac{5x+1}{2-5x}The numerator of f^{-1}(x) is Answer - AnswerThe denominator of f^{-1}(x) is Answer(Answer + Answer)

Answer 1

`\begin{gathered} \text{ The numerator of f}^(-1)(x)\text{ is 1-2x} \\ \text{ The denominator of f}^(-1)(x)\text{ is -5(x}+1) \end{gathered}`

Explanation:

To find the inverse of a function, replace f(x) by ''y'', then replace ''y'' with and x, and every x with a ''y''. Solve for y.

`\begin{gathered} f(x)=(5x+1)/(2-5x) \\ Replace\colon\text{ f(x)}\rightarrow y \\ y=(5x+1)/(2-5x) \\ Replace\colon\text{ y}\rightarrow x\text{ x}\rightarrow y \\ x=(5y+1)/(2-5y) \\ \text{ Solve for y.} \\ x(2-5y)=5y+1 \\ 2x-5yx=5y+1 \\ -5yx=5y+1-2x \\ -5yx-5y=1-2x \\ y(-5x-5)=1-2x \\ y=-(1-2x)/(5x+5) \\ y=-(1-2x)/(5(x+1)) \end{gathered}`
`\begin{gathered} Replacey\colon f^(-1)(x) \\ f^(-1)(x)=-(1-2x)/(5(x+1)) \end{gathered}`