So , Inverse of [tex]f(x) = \frac{x-7}{x}[/tex] is [tex]x = \frac{-7}{y-1}[/tex].
Step-by-step explanation:
An inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., f(x) = y if and only if g(y) = x. As an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. In functional notation this inverse function would be given by,
[tex]{\displaystyle g(y)={\frac {y+7}{5}}}[/tex]
With y = 5x − 7 we have that f(x) = y and g(y) = x. Here we have [tex]f(x) = \frac{x-7}{x}[/tex]
[tex]f(x) = \frac{x-7}{x}[/tex]
⇒ [tex]f(x) = \frac{x-7}{x}[/tex]
⇒ [tex]f(x) =1- \frac{7}{x}[/tex]
Let f(x) = y:
⇒ [tex]f(x) =1- \frac{7}{x}[/tex]
⇒ [tex]y = 1 - \frac{7}{x}[/tex]
⇒ [tex]y -1 = \frac{-7}{x}[/tex]
⇒ [tex]x = \frac{-7}{y-1}[/tex]
So , Inverse of [tex]f(x) = \frac{x-7}{x}[/tex] is [tex]x = \frac{-7}{y-1}[/tex].
