We are tasked with analyzing the function:

$f(x) = \frac{5}{2x - 3}$

Features of the graph:

1. Horizontal Asymptote:

The horizontal asymptote depends on the degrees of the numerator and denominator.

• The numerator (constant: 5) is of degree 0.
• The denominator ($2x - 3$) is of degree 1.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:

$y = 0$

2. Vertical Asymptote:

The vertical asymptote occurs where the denominator equals zero.

$2x - 3 = 0 \quad \implies \quad x = \frac{3}{2}$

Thus, the vertical asymptote is:

$x = \frac{3}{2}$

3. x-Intercept:

The $x$-intercept occurs when $f(x) = 0$, which means setting the numerator equal to zero:

$5 = 0$

Since this is never true, there is no $x$-intercept.

4. y-Intercept:

The $y$-intercept occurs when $x = 0$:

$f(0) = \frac{5}{2(0) - 3} = \frac{5}{-3} = -\frac{5}{3}$

Thus, the $y$-intercept is:

$(0, -\frac{5}{3})$

5. Hole:

A hole occurs if there are common factors in the numerator and denominator that cancel out. Since the numerator is a constant and the denominator does not share a common factor, there are no holes in the graph.

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Final Answer:

• Horizontal Asymptote: $y = 0$
• Vertical Asymptote: $x = \frac{3}{2}$
• x-Intercept: None
• y-Intercept: $(0, -\frac{5}{3})$
• Hole: None

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Would you like a graph or additional explanation for any step?

Related Questions:

1. How do you determine horizontal asymptotes for other rational functions?
2. Can a rational function have multiple vertical asymptotes?
3. What happens when the degree of the numerator equals the degree of the denominator?
4. How do holes in a graph relate to the domain of a function?
5. How does the horizontal asymptote influence end behavior?

Tip:

For rational functions, always check for common factors between numerator and denominator before identifying asymptotes and holes!