Question

consider the function $f(x) = \frac{c}{x}$, where $c$ is a nonzero real number. the vertical asymptote is $x = 0$. the horizontal asymptote is $y = 0$. the domain dropdown with options: cannot be determined, is all nonzero real numbers, is all real numbers except $c$, is all real numbers. the range dropdown with same options as domain.

Explanation:

Step1: Recall Domain of Rational Function

For \( f(x)=\frac{c}{x} \) ( \( c
eq0 \) ), the denominator \( x \) cannot be \( 0 \) (division by zero is undefined). So \( x \) can be any real number except \( 0 \). Wait, but the dropdown here is for domain? Wait, no, the dropdown shown is for domain? Wait, the function is \( f(x)=\frac{c}{x} \), domain is all real numbers except \( x = 0 \), but looking at the dropdown options (wait, the user's screenshot: the domain dropdown? Wait, no, the options given in the dropdown (the ones under the domain? Wait, no, the screenshot shows "The domain" dropdown with options: "cannot be determined", "is all nonzero real numbers", "is all real numbers except c", "is all real numbers". Wait, no, maybe I misread. Wait, the function is \( f(x)=\frac{c}{x} \), domain: \( x
eq0 \), but the options here—wait, maybe it's a typo, or maybe the question is about range? Wait, no, the label is "The domain" but the options seem off. Wait, no, let's re-express: \( f(x)=\frac{c}{x} \), so \( x \) can be any real number except \( 0 \), so domain is \( \mathbb{R}\setminus\{0\} \), but the options given: "is all nonzero real numbers"—wait, nonzero real numbers are \( x
eq0 \), so that's correct. Wait, but let's check again. Wait, the function is \( f(x)=\frac{c}{x} \), so \( x \) must be non - zero (since \( x = 0 \) makes denominator zero). So the domain is all real numbers except \( 0 \), which is "is all nonzero real numbers". Wait, but let's confirm: domain is the set of all possible \( x \) values. For \( f(x)=\frac{c}{x} \), \( x
eq0 \), so \( x \) is all nonzero real numbers. So the correct option for domain (if the dropdown is for domain) is "is all nonzero real numbers". Wait, but maybe the user made a mistake, but according to the function \( f(x)=\frac{c}{x} \), domain is \( x
eq0 \), so the domain is all nonzero real numbers.

Step2: Select Correct Option

From the dropdown options, "is all nonzero real numbers" matches the domain of \( f(x)=\frac{c}{x} \) (since \( x \) can't be \( 0 \), so \( x \) is any nonzero real number).

Answer:

is all nonzero real numbers