Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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].
### 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].
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
