Certainly! Let's analyze the given function [tex]\( f(x) = \frac{c}{x} \)[/tex], where [tex]\( c \)[/tex] is a nonzero real number. We will determine the vertical asymptote, the horizontal asymptote, the domain, and the range of this function.
### Vertical Asymptote
For the function [tex]\( f(x) = \frac{c}{x} \)[/tex], the vertical asymptote occurs where the function is undefined. Since division by zero is undefined, this happens when [tex]\( x = 0 \)[/tex]. Therefore, the vertical asymptote is:
[tex]\[ \text{Vertical asymptote: } x = 0 \][/tex]
### Horizontal Asymptote
To determine the horizontal asymptote, we look at the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity. For large values of [tex]\( x \)[/tex] in either direction, the term [tex]\( \frac{c}{x} \)[/tex] approaches 0 because [tex]\( c \)[/tex] is a constant and [tex]\( x \)[/tex] grows without bound. Therefore, the horizontal asymptote is:
[tex]\[ \text{Horizontal asymptote: } y = 0 \][/tex]
### Domain
The domain of the function is all the possible values of [tex]\( x \)[/tex] for which the function is defined. Since the function [tex]\( f(x) = \frac{c}{x} \)[/tex] is undefined when [tex]\( x = 0 \)[/tex], the domain is all real numbers except zero. Hence, the domain is:
[tex]\[ \text{Domain: } \text{all real numbers except } 0 \][/tex]
### Range
The range of the function is all the possible values of [tex]\( f(x) \)[/tex]. Since [tex]\( c \)[/tex] is a nonzero real number, [tex]\( \frac{c}{x} \)[/tex] can take any real value except zero as [tex]\( x \)[/tex] varies over all real numbers except 0.
For positive [tex]\( c \)[/tex]:
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( \frac{c}{x} \)[/tex] approaches 0 from the positive side.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( \frac{c}{x} \)[/tex] approaches 0 from the negative side.
For negative [tex]\( c \)[/tex]:
- As [tex]\( x \)[/tex] approaches infinity, [tex]\( \frac{-c}{x} \)[/tex] approaches 0 from the negative side.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( \frac{-c}{x} \)[/tex] approaches 0 from the positive side.
In both cases, [tex]\( f(x) \)[/tex] can take any real value except 0. Therefore, the range is:
[tex]\[ \text{Range: } \text{all real numbers except } 0 \][/tex]
### Summary
Putting it all together:
- The vertical asymptote is [tex]\( x = 0 \)[/tex].
- The horizontal asymptote is [tex]\( y = 0 \)[/tex].
- The domain is all real numbers except 0.
- The range is all real numbers except 0.
Thus, we have:
[tex]\[
\begin{aligned}
\text{Vertical asymptote:} & \quad x = 0 \\
\text{Horizontal asymptote:} & \quad y = 0 \\
\text{Domain:} & \quad \text{all real numbers except } 0 \\
\text{Range:} & \quad \text{all real numbers except } 0
\end{aligned}
\][/tex]
