Question

a. graph the function ( f(x)=e^{x} ) on the window ( -5,5 ) by ( -10,30 ).
b. find ( f(1), f(-3) ), and ( f(2) ), rounded to three decimal places.
c. what is the horizontal asymptote of the graph?
d. what is the ( y )-intercept?
a. choose the correct graph below.
b. ( f(1)=square ) (round to three decimal places as needed.)

Explanation:

Step1: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x)=e^{x} \).
\( f(1)=e^{1}\approx2.718 \)

Step2: Calculate \( f(-3) \)

Substitute \( x=-3 \) into \( f(x)=e^{x} \).
\( f(-3)=e^{-3}=\frac{1}{e^{3}}\approx\frac{1}{20.086}\approx0.050 \)

Step3: Calculate \( f(2) \)

Substitute \( x = 2 \) into \( f(x)=e^{x} \).
\( f(2)=e^{2}\approx7.389 \)

Step4: Find the horizontal asymptote

For \( y = e^{x} \), as \( x\to-\infty \), \( y\to0 \). So the horizontal asymptote is \( y = 0 \).

Step5: Find the \( y \)-intercept

Set \( x = 0 \) in \( f(x)=e^{x} \). Then \( f(0)=e^{0}=1 \). So the \( y \)-intercept is \( (0,1) \).

Answer:

• \( f(1)\approx2.718 \)
• \( f(-3)\approx0.050 \)
• \( f(2)\approx7.389 \)
• Horizontal asymptote: \( y = 0 \)
• \( y \)-intercept: \( (0,1) \)