Question

Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

Answer 1

Injectivity of f :
Let x and y be two elements of domain (R), such that
f(x) = f(y)
$\Rightarrow$4x + 3 = 4y + 3
$\Rightarrow$4x = 4y
$\Rightarrow$x = y
So, f is one-one.

Surjectivity of f :
Let y be in the co-domain (R), such that f(x) = y.

$$\begin{gathered} \Rightarrow 4x+3=y \\ \Rightarrow 4x=y-3 \\ \Rightarrow x=\frac{y-3}{4}\in R\left(\text{Domain}\right) \end{gathered}$$

$\Rightarrow$f is onto.
So, f is a bijection and, hence, is invertible.

Finding f ^-1:
$$\begin{gathered} \text{Let}{f}^{-1}\left(x\right)=y...\left(1\right) \\ \Rightarrow x=f\left(y\right) \\ \Rightarrow x=4y+3 \\ \Rightarrow x-3=4y \\ \Rightarrow y=\frac{x-3}{4} \\ \text{So,}{f}^{-1}\left(x\right)=\frac{x-3}{4}[\text{from}\left(1\right)] \end{gathered}$$