Question

select all the correct answers. consider the parent function $f(x)=e^x$ and transformed function $g(x)=-f(x)-4$. which features of function $f$ and function $g$ are different? - domain - y-intercept - range - horizontal asymptote - end behavior

Explanation:

Step1: Analyze domain of f(x)

The parent function $f(x)=e^x$ has a domain of all real numbers, $(-\infty, \infty)$.

Step2: Analyze domain of g(x)

The transformed function $g(x)=-f(x)-4=-e^x-4$ also accepts all real numbers as inputs, so its domain is $(-\infty, \infty)$.

Step3: Calculate y-intercept of f(x)

Set $x=0$: $f(0)=e^0=1$.

Step4: Calculate y-intercept of g(x)

Set $x=0$: $g(0)=-e^0-4=-1-4=-5$.

Step5: Analyze range of f(x)

For $f(x)=e^x$, $e^x>0$ for all real $x$, so range is $(0, \infty)$.

Step6: Analyze range of g(x)

For $g(x)=-e^x-4$, since $e^x>0$, then $-e^x<0$, so $-e^x-4<-4$. Range is $(-\infty, -4)$.

Step7: Analyze horizontal asymptote of f(x)

As $x\to-\infty$, $e^x\to0$, so horizontal asymptote is $y=0$.

Step8: Analyze horizontal asymptote of g(x)

As $x\to-\infty$, $-e^x-4\to0-4=-4$, so horizontal asymptote is $y=-4$.

Step9: Analyze end behavior of f(x)

As $x\to\infty$, $e^x\to\infty$; as $x\to-\infty$, $e^x\to0$.

Step10: Analyze end behavior of g(x)

As $x\to\infty$, $-e^x-4\to-\infty$; as $x\to-\infty$, $-e^x-4\to-4$.

Answer:

• y-intercept
• range
• horizontal asymptote
• end behavior