Let's correct the given analysis for the function [tex]\( f(x) = \frac{c}{x} \)[/tex], where [tex]\( c \)[/tex] is a nonzero real number. We'll find the actual vertical asymptote, horizontal asymptote, domain, and range, one step at a time.
### Vertical Asymptote:
The function [tex]\( f(x) = \frac{c}{x} \)[/tex] becomes undefined where the denominator is zero. Therefore, [tex]\( x = 0 \)[/tex] is a point where the function is undefined and is thus the vertical asymptote.
Vertical Asymptote: [tex]\( x = 0 \)[/tex]
### Horizontal Asymptote:
To determine the horizontal asymptote, we consider the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( +\infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]). For large values of [tex]\( x \)[/tex], both positive and negative, [tex]\( f(x) \)[/tex] approaches zero since the numerator [tex]\( c \)[/tex] is a constant and the denominator [tex]\( x \)[/tex] becomes very large.
Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
### Domain:
The function [tex]\( f(x) = \frac{c}{x} \)[/tex] is defined for all real numbers except where the denominator is zero. Therefore, the domain includes all real numbers except [tex]\( x = 0 \)[/tex].
Domain: [tex]\( (-\infty, 0) \cup (0, +\infty) \)[/tex]
### Range:
The range of [tex]\( f(x) = \frac{c}{x} \)[/tex] consists of all values that [tex]\( f(x) \)[/tex] can take. As [tex]\( x \)[/tex] approaches positive and negative infinity, [tex]\( f(x) \)[/tex] approaches zero from both positive and negative directions but never actually equals zero. Thus, the range excludes zero.
Range: [tex]\( (-\infty, 0) \cup (0, +\infty) \)[/tex]
Corrected Results:
- Vertical Asymptote: [tex]\( x = 0 \)[/tex]
- Horizontal Asymptote: [tex]\( y = 0 \)[/tex]
- Domain: [tex]\( (-\infty, 0) \cup (0, +\infty) \)[/tex]
- Range: [tex]\( (-\infty, 0) \cup (0, +\infty) \)[/tex]
These corrected results reflect the true characteristics of the function [tex]\( f(x) = \frac{c}{x} \)[/tex].
