select all the correct answers. consider the parent function f(x) = e^x and transformed function g(x) =…

Explanation:

Step1: Analyze the parent - function $f(x)=e^{x}$

The domain of $y = e^{x}$ is $(-\infty,\infty)$ since we can input any real - number for $x$. The range is $(0,\infty)$ because $e^{x}>0$ for all real $x$. The horizontal asymptote is $y = 0$ as $x\to-\infty$, $e^{x}\to0$. The $y$ - intercept is found by setting $x = 0$, so $y=e^{0}=1$. As $x\to\infty$, $e^{x}\to\infty$ and as $x\to-\infty$, $e^{x}\to0$.

Step2: Analyze the transformed function $g(x)=-f(x)-4=-e^{x}-4$

The domain of $g(x)$ is also $(-\infty,\infty)$ because we can input any real number for $x$. To find the range, since $e^{x}>0$, then $-e^{x}<0$ and $-e^{x}-4 < - 4$, so the range of $g(x)$ is $(-\infty,-4)$. The horizontal asymptote: as $x\to-\infty$, $-e^{x}\to0$ and $g(x)\to - 4$, so the horizontal asymptote is $y=-4$. The $y$ - intercept: set $x = 0$, then $g(0)=-e^{0}-4=-1 - 4=-5$. For the end - behavior, as $x\to\infty$, $-e^{x}\to-\infty$ and $g(x)\to-\infty$; as $x\to-\infty$, $-e^{x}\to0$ and $g(x)\to - 4$.

Step3: Compare features

• Domain: Both $f(x)$ and $g(x)$ have a domain of $(-\infty,\infty)$, so the domain is the same.
• Range: The range of $f(x)$ is $(0,\infty)$ and the range of $g(x)$ is $(-\infty,-4)$, so the range is different.
• Horizontal asymptote: The horizontal asymptote of $f(x)$ is $y = 0$ and of $g(x)$ is $y=-4$, so the horizontal asymptote is different.
• $y$ - intercept: The $y$ - intercept of $f(x)$ is $1$ and of $g(x)$ is $-5$, so the $y$ - intercept is different.
• End - behavior: As $x\to\infty$, $f(x)\to\infty$ and $g(x)\to-\infty$; as $x\to-\infty$, $f(x)\to0$ and $g(x)\to - 4$, so the end - behavior is different.

Answer:

horizontal asymptote, range, y - intercept, end behavior