Solution assignment 04 Fractional functions and graphs

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Assignment 4

Given the function:

$y=1+\displaystyle\frac{2}{x-3}$

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 of this function is $x=3$ . We find the horizontal asymptote by taking $x\rightarrow\infty$ or $x\rightarrow-\infty$ . We get $y=1$ because the second term approaches $0$ and then $y$ approaches $1$ . The intersection point with the $Y$ -axis is found by substituting $x=0$ in the formula. We get the intersection point with the $X$ -axis by substituting $y=0$ and solving the equation: