3. Draw the graph of the function $f$ defined by $f(x)=-\frac{1}{2}+\frac{7}{2}(2 x+3)^{-1}$. Which line (write its equation) does not intersect the graph of the function $f$ and is parallel to the $x$-axis?

(7 points)
3. We simplify the function rule $f: f(x)=-\frac{1}{2}+\frac{7}{2(2 x+3)}=\frac{-2 x+4}{2(2 x+3)}=\frac{2-x}{2 x+3}$. We calculate the zero $x=2$, the pole $x=-\frac{3}{2}$, the horizontal asymptote $y=-\frac{1}{2}$ of the function $f$, and the coordinates of the intersection of the graph with the y-axis $N\left(0, \frac{2}{3}\right)$. We draw the graph. The line that is parallel to the $x$-axis and does not intersect the graph is the horizontal asymptote $y=-\frac{1}{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-14.jpg?height=771&width=780&top_left_y=200&top_left_x=638)

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-14.jpg?height=291&width=1644&top_left_y=1008&top_left_x=203)