To solve the problem for the function [tex]\( f(x) = \frac{1}{(x-4)^2} \)[/tex], we'll break it down step-by-step.
### 1. Domain
The domain of [tex]\( f(x) \)[/tex] consists of all the x-values for which the function is defined. Since [tex]\( f(x) \)[/tex] involves a denominator [tex]\((x-4)^2\)[/tex], the function becomes undefined when [tex]\( x = 4 \)[/tex]:
[tex]\[ \text{Domain: } (-\infty, 4) \cup (4, \infty) \][/tex]
### 2. x- and y-Intercepts
#### x-intercept:
To find the x-intercept, we set [tex]\( f(x) \)[/tex] to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \frac{1}{(x-4)^2} \][/tex]
This equation has no solution because the numerator is always 1, and [tex]\((x-4)^2\)[/tex] is never zero. Therefore, there is no x-intercept.
#### y-intercept:
To find the y-intercept, we set [tex]\( x = 0 \)[/tex] in the function and solve for [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = \frac{1}{(0-4)^2} = \frac{1}{16} \][/tex]
So, the y-intercept is [tex]\( \left( 0, \frac{1}{16} \right) \)[/tex].
### 3. Vertical Asymptote
A vertical asymptote occurs where the function goes to infinity. This happens when the denominator equals zero. Hence, [tex]\( x = 4 \)[/tex] is a vertical asymptote.
### 4. Horizontal Asymptotes
To determine horizontal asymptotes, we look at the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]:
[tex]\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{1}{(x-4)^2} = 0 \][/tex]
[tex]\[ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \frac{1}{(x-4)^2} = 0 \][/tex]
So, the horizontal asymptote is:
[tex]\[ y = 0 \][/tex]
### 5. Oblique Asymptotes
Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. In this case, the numerator is 1 (degree 0) and the denominator is [tex]\((x-4)^2\)[/tex] (degree 2), which means there is no oblique asymptote.
### 6. Graph
The correct graph of [tex]\( f(x) = \frac{1}{(x-4)^2} \)[/tex] should have:
- A vertical asymptote at [tex]\( x = 4 \)[/tex]
- A horizontal asymptote at [tex]\( y = 0 \)[/tex]
- No x-intercept
- A y-intercept at [tex]\( \left( 0, \frac{1}{16} \right) \)[/tex]
Given no graphs to choose from, I'll describe the general shape: The graph will have two branches, each approaching the vertical asymptote at [tex]\( x = 4 \)[/tex] as they go to infinity. Both branches will also approach the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] goes to positive or negative infinity and will not cross the x-axis.
### Summary:
1. Domain: [tex]\( (-\infty, 4) \cup (4, \infty) \)[/tex]
2. x-intercept: None
3. y-intercept: [tex]\( \left( 0, \frac{1}{16} \right) \)[/tex]
4. Vertical Asymptote: [tex]\( x = 4 \)[/tex]
5. Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
6. Oblique Asymptote: None
