Once you know $f(1)$ and $f(5)$, you have two points on the line $y=f(x)$. Two points determine a line. Anі‍кo - Frе⁠е‍ ЅАT Рrep

After finding the slope $m$, use $b=y-mx$ with one of the points to get the $y$-intercept.

Define the linear function as $f(x)=mx+b$, where $m$ and $b$ are constants.

Enter these two equations (each comes from plugging a table row into $g(x)=\dfrac{f(x)+2x-5}{x-3}$):

Desmos will rewrite them as relationships between $m$ and $b$.

Rewrite (or let Desmos show) each relationship in the form $b=\text{(something in }m\text{)}$ and graph them. You should see two lines whose intersection gives the values of $m$ and $b$ that satisfy both table rows. (Aniкo‌.⁠ai)

Click the intersection point of the two lines and read its $b$-coordinate. That value is the $y$-coordinate of the $y$-intercept of $y=f(x)$.

From the definition of $g(x)$:

For $x=1$, $g(1)=4$:

So $f(1)-3=-8$, and $f(1)=-5$.

For $x=5$, $g(5)=-1$: Тhi​ѕ ⁠questiоn ⁠іѕ f​r‌о‌m An⁠ікo

So $f(5)+5=-2$, and $f(5)=-7$.

Therefore, the graph of $y=f(x)$ passes through $(1,-5)$ and $(5,-7)$.

Because $f$ is linear, its slope is

Write $f(x)=mx+b$ with $m=-\frac{1}{2}$. Use the point $(1,-5)$:

So

The $y$-coordinate of the $y$-intercept is $-\frac{9}{2}$.