Let's analyze the function [tex]\( f(x) = \frac{c}{x} \)[/tex] where [tex]\( c \)[/tex] is a nonzero real number.
### Vertical Asymptote:
A vertical asymptote occurs at values of [tex]\( x \)[/tex] where the function tends toward [tex]\(\pm\infty\)[/tex]. For the function [tex]\( f(x) = \frac{c}{x} \)[/tex], the denominator is zero at [tex]\( x = 0 \)[/tex]. Therefore, the function has a vertical asymptote at [tex]\( x = 0 \)[/tex]. As [tex]\( x \)[/tex] approaches 0 from the left and the right, the function values behave differently depending on the sign of [tex]\( c \)[/tex]:
[tex]\[ \lim_{{x \to 0^-}} \frac{c}{x} = -\infty \cdot \text{sign}(c) \][/tex]
[tex]\[ \lim_{{x \to 0^+}} \frac{c}{x} = \infty \cdot \text{sign}(c) \][/tex]
This means the function tends to negative infinity if [tex]\( c \)[/tex] is positive (or positive infinity if [tex]\( c \)[/tex] is negative) when approaching zero from the left, and to positive infinity if [tex]\( c \)[/tex] is positive (or negative infinity if [tex]\( c \)[/tex] is negative) when approaching from the right.
### Horizontal Asymptote:
A horizontal asymptote describes the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(\pm\infty\)[/tex]. Examining the limits:
[tex]\[ \lim_{{x \to \infty}} \frac{c}{x} = 0 \][/tex]
[tex]\[ \lim_{{x \to -\infty}} \frac{c}{x} = 0 \][/tex]
Therefore, the function has a horizontal asymptote at [tex]\( y = 0 \)[/tex].
### Domain:
The domain of the function [tex]\( f(x) = \frac{c}{x} \)[/tex] is all real numbers except where the denominator is zero. Since the denominator [tex]\( x \)[/tex] is zero at [tex]\( x = 0 \)[/tex], the function is undefined at that point. Thus, the domain is all real numbers except zero:
[tex]\[ \text{Domain} = \mathbb{R} - \{0\} \][/tex]
Or in interval notation:
[tex]\[ \text{Domain} = (-\infty, 0) \cup (0, \infty) \][/tex]
### Range:
The range of the function [tex]\( f(x) = \frac{c}{x} \)[/tex] can be determined by considering the values that [tex]\( f(x) \)[/tex] can take. For any nonzero real number [tex]\( c \)[/tex], and for all [tex]\( x \neq 0 \)[/tex], the function [tex]\( f(x) \)[/tex] can take any real value except for zero. Therefore, the range is all real numbers except zero:
[tex]\[ \text{Range} = \mathbb{R} - \{0\} \][/tex]
Or in interval notation:
[tex]\[ \text{Range} = (-\infty, 0) \cup (0, \infty) \][/tex]
### Conclusion:
- The vertical asymptote is at [tex]\( x = 0 \)[/tex].
- The horizontal asymptote is [tex]\( y = 0 \)[/tex].
- The domain is [tex]\( (-\infty, 0) \cup (0, \infty) \)[/tex].
- The range is [tex]\( (-\infty, 0) \cup (0, \infty) \)[/tex].
