Question

consider the function $g(x) = \frac{10}{x}$. the vertical asymptote is $x = \boxed{0}$ \checkmark. the horizontal asymptote is $y = \boxed{0}$ \checkmark. complete select all that are true for $g(x) = \frac{10}{x}$. $\square$ the domain of $g(x)$ is all real numbers. $\square$ the domain of $g(x)$ is the same as the domain of the parent function. $\square$ the range of $g(x)$ is all real numbers. $\square$ the range is the same as the range of the parent function. $\square$ the function $g(x)$ increases over the same $x$-values as the parent function. $\square$ the function $g(x)$ decreases over the same $x$-values as the parent function. done \checkmark

Explanation:

• Domain: For \(g(x)=\frac{10}{x}\), the denominator cannot be zero. The parent function of \(y = \frac{10}{x}\) is \(y=\frac{1}{x}\). The domain of \(y=\frac{10}{x}\) and \(y = \frac{1}{x}\) is \(x

eq0\), so the domain of \(g(x)\) is not all real numbers (since \(x = 0\) is excluded), but it is the same as the domain of the parent function \(y=\frac{1}{x}\).

• Range: For \(y=\frac{10}{x}\), as \(x\to\pm\infty\), \(y\to0\) but \(y

eq0\). The parent function \(y=\frac{1}{x}\) has a range of \(y
eq0\). So the range of \(g(x)\) is not all real numbers, but it is the same as the range of the parent function.

• Increasing/Decreasing: The function \(y=\frac{10}{x}\) and \(y=\frac{1}{x}\) are decreasing on \((-\infty,0)\) and \((0,\infty)\) (using the derivative \(y'=-\frac{10}{x^{2}}<0\) for \(x

eq0\) and \(y'=-\frac{1}{x^{2}}<0\) for \(x
eq0\) for the parent function). They do not increase on any interval.

Answer:

• B. The domain of \(g(x)\) is the same as the domain of the parent function.
• D. The range is the same as the range of the parent function.
• F. The function \(g(x)\) decreases over the same \(x -\)values as the parent function.