The problem can be solved in multiple ways. We solve it here by first computing the distribution function $G(y)$ of $Y$.

\begin{align}G(y) &=P(Y\leq y)\\
&=P(cX+d\leq y)\\
&=P\left(X\leq\frac{y-d}{c}\right)\\
&=F\left(\frac{y-d}{c}\right)\\
\end{align}

Note above that the inequality does not change its direction since $c>0$. Now, the density function of $Y$

\begin{align}g(y) &=\frac{d}{dy}G(y)\\
&=\frac{d}{dy}F\left(\frac{y-d}{c}\right)\\
&=F'\left(\frac{y-d}{c}\right)\cdot\frac{d}{dy}\left(\frac{y-d}{c}\right),\mbox{ chain rule}\\
&=\begin{cases}f\left(\frac{y-d}{c}\right)\cdot\frac{1}{c},\mbox{ where } a\leq\frac{y-d}{c}\leq b\\
0,\mbox{ otherwise}\end{cases}\\
&=\begin{cases}\frac{1}{c}f\left(\frac{y-d}{c}\right),\mbox{ where } ac+d\leq y\leq bc+d\\
0,\mbox{ otherwise}\end{cases}\\
\end{align}