Answer
$\color{blue}{f_X(x) = x/21, x=0,\ldots, 6}.$
Work Step by Step
For $X=0$,
$\begin{align*}
f_X(0) &= P(X\le 0) - P(X\lt 0) \\
&= F(0) - 0 & [\text{since}\ F_X(x)\ne 0,\ \text{only for}\ x=0,1,\ldots, 6] \\
&= \frac{0(0+1)}{42} \\
f_X(0) &= 0.
\end{align*}$
For $X=1,\ldots, 6,$
$\begin{align*}
f_X(x) &= P(X=x) \\
&= P(X\le x) - P(X\lt x),\ x=1,\ldots,6 \\
&= P(X\le x) - P(X\le x-1) & [\text{since}\ F_X(x)\ne 0,\text{only for}\ x=0,1,\ldots, 6] \\
&= F(x) -F(x-1) \\
&= (x)(x+1)/42 - (x-1)(x)/42 \\
&= (x)((x+1)-(x-1))/42 \\
&= (x)(2)/42 \\
f_X(x) &= x/21
\end{align*}$
Since $f_X(0) = 0/21 =0 ,$ we can then define $\color{blue}{f_X(x) = x/21, x=0,\ldots, 6}.$
