Answer to Question #191288 in Real Analysis for Parul

Question

Show that the function f :R -> R defined by f(x) = 2x+ 7 has an inverse by applying the inverse function theorem. Find its inverse also 

Answer 1

We have given the function,

$f: R \rightarrow R$ defined by $f(x) = 2x+7$

Since, the given function is bijective because for every y there exist a unique x

$x = \dfrac{y-7}{7}$ , such that $f(x) = y$

In general we can say that $R \rightarrow R$

$f(x) = ax+b, a \ne 0$

Hence, we can say that inverse exist for the given function $f(x)$.

Calculation of $f^{-1}(x).$

$f(x)= 2x+7$

$y = 2x+7$

$x = \dfrac{y-7}{2}$

where, $x = f^{-1}(x).$

Hence, $f^{-1}(x) = \dfrac{y-7}{2}.$