Question

$f:R\to R$given by f$\left(x\right)=4x+3$, show that $f$ is invertable. Find the inverse of $f$.

Answer 1

Solution :-

$f:R\to R$ given by $f\left(x\right)=4x+3$

To show f is invertible, we need to show it is one-one

and onto.

Checking one-one

$f\left(x_1\right)=4x_1+3$

$f\left(x_2\right)=4x_2+3$

putting $f\left(x_1\right)=f\left(x_2\right)$

$\Rightarrow 4x_1+3=4x_2+3$

$\Rightarrow x_1=x_2$

If $f\left(x_1\right)-f\left(x_2\right)$, then $x_1=x_2$

$\therefore$ f is one-one

Checking onto

$f\left(x\right)=4x+3$

Let f(x) = y. where $g\in y$

$y=4x+3$

$y-3=4x$

$\Rightarrow x=\frac{y-3}{4},$ $y$ is real no , so $\frac{y-3}{4}$ is real

$\therefore$ $f$ is onto, Hence $f$ is invertible

To find inverse, we put $f\left(x\right)=y$

$\Rightarrow y=4x+3\Rightarrow x=\frac{y-3}{4}$

$\lambda =f^{-1}(4x+3)=\frac{y-3}{4}$