Answer:
[tex]\boxed{\pink{\sf The \ inverse\ of \ the \ function \ is \ f^{-1}(x)= \dfrac{2}{x}-\dfrac{1}{2} }}[/tex]
Step-by-step explanation:
A function is given to us and we need to find its inverse . So the given function is ,
[tex]\implies f(x) = \dfrac{4}{2x+1} [/tex]
1) So ,let us take f(x) = y . Equation becomes ,
[tex]\implies y = \dfrac{4}{2x+1} [/tex]
2) Firstly replace x with y and y with x. We get ,
[tex]\implies x = \dfrac{4}{2y+1} [/tex]
3) Now , solve for y.
[tex]\implies x = \dfrac{4}{2y+1} \\\\\implies x(2y + 1) = 4 \\\\\implies 2y+1 =\dfrac{4}{x}\\\\\implies 2y = \dfrac{4}{x}-1 \\\\\implies 2y = \dfrac{4-x}{x} \\\\\implies y = \dfrac{4-x}{2x} [/tex]
4) Now replace y with [tex]f^{-1}(x) [/tex]
[tex]\implies f^{-1}(x) = \dfrac{4-x}{2x} \\\\\ \implies f^{-1}(x)= \dfrac{4}{2x} - \dfrac{x}{2x} \\\\\underline{\boxed{\orange{\tt \implies f^{-1}(x) = \dfrac{2}{x}-\dfrac{1}{2} }}}[/tex]
