Alright, let's determine each feature of the given function [tex]\( f(x) = \frac{4}{2x - 3} \)[/tex] step by step.
### Horizontal Asymptote
To find the horizontal asymptote of the function, we need to analyze the degrees of the polynomials in the numerator and the denominator.
- The numerator is a constant [tex]\(4\)[/tex] (degree 0).
- The denominator is [tex]\(2x - 3\)[/tex] (degree 1).
Since the degree of the numerator (0) is less than the degree of the denominator (1), the horizontal asymptote is [tex]\(y = 0\)[/tex].
Horizontal Asymptote:
[tex]\[ y = 0 \][/tex]
### Vertical Asymptote
To find the vertical asymptote, we set the denominator equal to zero and solve for [tex]\(x\)[/tex].
[tex]\[ 2x - 3 = 0 \][/tex]
[tex]\[ 2x = 3 \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
So, the vertical asymptote is [tex]\(x = 1.5\)[/tex].
Vertical Asymptote:
[tex]\[ x = \frac{3}{2} \][/tex]
[tex]\[ x = 1.5 \][/tex]
### [tex]\(x\)[/tex]-Intercept
To find the [tex]\(x\)[/tex]-intercept, we set the numerator equal to zero and solve for [tex]\(x\)[/tex].
[tex]\[ 4 = 0 \][/tex]
Since this equation is never true, there is no [tex]\(x\)[/tex]-intercept.
[tex]\(x\)[/tex]-Intercept:
[tex]\[ \text{None} \][/tex]
### [tex]\(y\)[/tex]-Intercept
To find the [tex]\(y\)[/tex]-intercept, we set [tex]\(x = 0\)[/tex] and solve for [tex]\(y\)[/tex].
[tex]\[ f(0) = \frac{4}{2 \cdot 0 - 3} = \frac{4}{-3} \][/tex]
[tex]\[ f(0) = -\frac{4}{3} \][/tex]
So, the [tex]\(y\)[/tex]-intercept is [tex]\((0, -\frac{4}{3})\)[/tex].
[tex]\(y\)[/tex]-Intercept:
[tex]\[ (0, -\frac{4}{3}) \][/tex]
[tex]\[ (0, -1.333 \, \text{approximately}) \][/tex]
### Hole
A hole in the graph occurs where both the numerator and denominator of a rational function are zero for the same value of [tex]\(x\)[/tex].
Since the function is defined for all [tex]\(x\)[/tex] except where the denominator is zero, and there is no common factor in the numerator and denominator that would cancel out, there are no holes in the graph of this function.
Hole:
[tex]\[ \text{None} \][/tex]
### Summary
Here are all the key features of the graph summarized:
- Horizontal Asymptote: [tex]\(y = 0\)[/tex]
- Vertical Asymptote: [tex]\(x = \frac{3}{2}\)[/tex] or [tex]\(x = 1.5\)[/tex]
- [tex]\(x\)[/tex]-Intercept: None
- [tex]\(y\)[/tex]-Intercept: [tex]\((0, -\frac{4}{3})\)[/tex] or [tex]\((0, -1.333)\)[/tex]
- Hole: None
