Solution assignment 03 Fractional functions and graphs

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Assignment 3

Given the function:

$y=-1+\displaystyle\frac{1}{x+3}$

Find:
the vertical asymptote;
the horizontal asymptote;
the intersection point with the $Y$ -as if it exists;
the intersection point with the $X$ -as if it exists.

Based on these results sketch the result in the figure.

Solution

The vertical asymptote is $x=-3$ . We find the horizontal asymptote when $x\rightarrow\infty$ or $x\rightarrow-\infty$ and that is $y=-1$ , because the second term approaches $0$ and thus $y$ approaches $-1$ . The intersection point with the $Y$ -axis is $y=-\displaystyle\frac{2}{3}$ . This result can be obtained by substituting $x=0$ in the function. The intersection point with the $X$ -as is obtained by substituting $y=0$ and solving the following equation: