Question

If $f:N\to Y$ is a function defined as $f\left(x\right)=4x+3$, where $Y=\left\{\forall \in N:y=4x+3\text{for some}x\in N\right\}$. Then, the inverse of $f$ is

• A. $\mathrm{g}\left(\mathrm{y}\right)=\frac{\left[y-3\right]}{4}$
• B. $\mathrm{g}\left(\mathrm{y}\right)=\frac{\left[3y+4\right]}{3}$
• C. $\mathrm{g}\left(\mathrm{y}\right)=4+\frac{\left[y+3\right]}{4}$
• D. $\mathrm{g}\left(\mathrm{y}\right)=\frac{\left[y+3\right]}{4}$

Answer 1

The correct option is A

$\mathrm{g}\left(\mathrm{y}\right)=\frac{\left[y-3\right]}{4}$

Explanation for the correct option

To find the inverse of a function we need to write $x$ in terms of $y$,

$\begin{array}{rcl}\therefore \mathrm{y}& =& 4\mathrm{x}+3\\ & \Rightarrow & \mathrm{y}-3=4\mathrm{x}\\ & \Rightarrow & \mathrm{x}=\frac{\mathrm{y}-3}{4}\end{array}$

Now, we can write that $y=f\left(x\right)$ and $x=g\left(y\right)$. Now this $g\left(y\right)$ is the inverse of $f$ because $f^{-1}\left(y\right)=g\left(y\right)$. From this, we have

$g\left(y\right)=\frac{y-3}{4}$

Hence, the correct option is option (A).