Answer :
Certainly! Let's provide a detailed step-by-step solution for the function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex].
### Step-by-Step Solution:
1. Understanding the Function:
The function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] is a rational function where the numerator is -1 and the denominator is [tex]\( x - 4 \)[/tex].
2. Identifying the Domain:
- The function [tex]\( f(x) \)[/tex] is defined for all [tex]\( x \)[/tex] except where the denominator is zero.
- To find where the denominator is zero, set [tex]\( x - 4 = 0 \)[/tex].
- Solving this, [tex]\( x = 4 \)[/tex].
- Therefore, the domain of the function is all real numbers except [tex]\( x = 4 \)[/tex]. In interval notation, this is [tex]\( (-\infty, 4) \cup (4, \infty) \)[/tex].
3. Analyzing Vertical Asymptote:
- A vertical asymptote occurs where the function's denominator is zero and the numerator is non-zero.
- Here, [tex]\( x = 4 \)[/tex] is the location of the vertical asymptote.
4. Determining Horizontal Asymptote:
- For large values of [tex]\( |x| \)[/tex], to determine the behavior of [tex]\( f(x) \)[/tex], we see that [tex]\( x - 4 \)[/tex] behaves like [tex]\( x \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
- Therefore, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
5. Graphing the Function:
- The function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] will approach the vertical asymptote [tex]\( x = 4 \)[/tex] but never touch or cross it.
- It will also approach the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] goes to positive or negative infinity.
6. Behavior Near the Asymptotes:
- As [tex]\( x \)[/tex] approaches 4 from the left ([tex]\( x \to 4^- \)[/tex]), [tex]\( x - 4 \)[/tex] is negative and very small; thus, [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] becomes positive and very large.
- As [tex]\( x \)[/tex] approaches 4 from the right ([tex]\( x \to 4^+ \)[/tex]), [tex]\( x - 4 \)[/tex] is positive and very small; thus, [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] becomes negative and very large in magnitude but negative in value.
### Intervals and Signs:
- For [tex]\( x < 4 \)[/tex]:
[tex]\( x - 4 \)[/tex] is negative, so [tex]\( -\frac{1}{x-4} \)[/tex] is positive.
- For [tex]\( x > 4 \)[/tex]:
[tex]\( x - 4 \)[/tex] is positive, so [tex]\( -\frac{1}{x-4} \)[/tex] is negative.
### Summary:
- Vertical Asymptote: [tex]\( x = 4 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- Domain: [tex]\( (-\infty, 4) \cup (4, \infty) \)[/tex]
- The graph approaches [tex]\( y = 0 \)[/tex] as [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex].
- The function value becomes very large positive as [tex]\( x \to 4^- \)[/tex] and very large negative as [tex]\( x \to 4^+ \)[/tex].
Thus, we have analyzed the function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] in great detail!
### Step-by-Step Solution:
1. Understanding the Function:
The function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] is a rational function where the numerator is -1 and the denominator is [tex]\( x - 4 \)[/tex].
2. Identifying the Domain:
- The function [tex]\( f(x) \)[/tex] is defined for all [tex]\( x \)[/tex] except where the denominator is zero.
- To find where the denominator is zero, set [tex]\( x - 4 = 0 \)[/tex].
- Solving this, [tex]\( x = 4 \)[/tex].
- Therefore, the domain of the function is all real numbers except [tex]\( x = 4 \)[/tex]. In interval notation, this is [tex]\( (-\infty, 4) \cup (4, \infty) \)[/tex].
3. Analyzing Vertical Asymptote:
- A vertical asymptote occurs where the function's denominator is zero and the numerator is non-zero.
- Here, [tex]\( x = 4 \)[/tex] is the location of the vertical asymptote.
4. Determining Horizontal Asymptote:
- For large values of [tex]\( |x| \)[/tex], to determine the behavior of [tex]\( f(x) \)[/tex], we see that [tex]\( x - 4 \)[/tex] behaves like [tex]\( x \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches 0.
- Therefore, the horizontal asymptote is [tex]\( y = 0 \)[/tex].
5. Graphing the Function:
- The function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] will approach the vertical asymptote [tex]\( x = 4 \)[/tex] but never touch or cross it.
- It will also approach the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] goes to positive or negative infinity.
6. Behavior Near the Asymptotes:
- As [tex]\( x \)[/tex] approaches 4 from the left ([tex]\( x \to 4^- \)[/tex]), [tex]\( x - 4 \)[/tex] is negative and very small; thus, [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] becomes positive and very large.
- As [tex]\( x \)[/tex] approaches 4 from the right ([tex]\( x \to 4^+ \)[/tex]), [tex]\( x - 4 \)[/tex] is positive and very small; thus, [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] becomes negative and very large in magnitude but negative in value.
### Intervals and Signs:
- For [tex]\( x < 4 \)[/tex]:
[tex]\( x - 4 \)[/tex] is negative, so [tex]\( -\frac{1}{x-4} \)[/tex] is positive.
- For [tex]\( x > 4 \)[/tex]:
[tex]\( x - 4 \)[/tex] is positive, so [tex]\( -\frac{1}{x-4} \)[/tex] is negative.
### Summary:
- Vertical Asymptote: [tex]\( x = 4 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- Domain: [tex]\( (-\infty, 4) \cup (4, \infty) \)[/tex]
- The graph approaches [tex]\( y = 0 \)[/tex] as [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex].
- The function value becomes very large positive as [tex]\( x \to 4^- \)[/tex] and very large negative as [tex]\( x \to 4^+ \)[/tex].
Thus, we have analyzed the function [tex]\( f(x) = -\frac{1}{x-4} \)[/tex] in great detail!
