The function given is $f(x) = \frac{x + 5}{x - 3}$. To sketch the graph of this function, let's break it down step by step:

1. Domain:

The function has a vertical asymptote where the denominator is zero, i.e., $x - 3 = 0$. Thus, the vertical asymptote occurs at: $x = 3$ The function is undefined at $x = 3$.

2. Horizontal Asymptote:

Since the degrees of the numerator and denominator are the same (both linear), the horizontal asymptote is determined by the ratio of the leading coefficients of $x$ in the numerator and denominator. Both coefficients are 1, so: $y = \frac{1}{1} = 1$ This gives us a horizontal asymptote at $y = 1$.

3. Intercepts:

• x-intercept: Set $f(x) = 0$ to find the x-intercept: $\frac{x + 5}{x - 3} = 0 \Rightarrow x + 5 = 0 \Rightarrow x = -5$ So the x-intercept is at $(-5, 0)$.

• y-intercept: To find the y-intercept, set $x = 0$: $f(0) = \frac{0 + 5}{0 - 3} = \frac{5}{-3} = -\frac{5}{3}$ So the y-intercept is at $\left(0, -\frac{5}{3}\right)$.

4. Behavior near the asymptotes:

• As $x$ approaches 3 from the left ($x \to 3^-$), $f(x)$ tends towards $-\infty$.
• As $x$ approaches 3 from the right ($x \to 3^+$), $f(x)$ tends towards $+\infty$.
• As $x \to \pm \infty$, $f(x) \to 1$, consistent with the horizontal asymptote.

5. Sketch:

• The graph will have a vertical asymptote at $x = 3$ and a horizontal asymptote at $y = 1$.
• The function passes through $(-5, 0)$ (x-intercept) and $\left(0, -\frac{5}{3}\right)$ (y-intercept).
• The function behaves like a rational function with opposite trends on either side of the vertical asymptote.

Would you like me to generate a graph for this function?

Relative Questions:

1. How can we determine the end behavior of rational functions?
2. What happens to the function as $x \to \infty$?
3. Can a rational function have more than one vertical asymptote?
4. How can the degrees of the numerator and denominator affect the graph's shape?
5. How do we find slant asymptotes in rational functions?

Tip:

Always check for both vertical and horizontal asymptotes in rational functions as they define the boundaries and behavior of the graph.