Question

construct a table of values for the function. (round your answers to three decimal places.)
$f(x) = 5e^{x} + 1$

$x$	$-3$	$-2$	$-1$	$0$	$1$
$f(x)$

sketch the graph of the function.
(images of graphs are shown)

Explanation:

Step1: For \( x = -3 \)

We substitute \( x = -3 \) into the function \( f(x)=5e^{x}+1 \). First, calculate \( e^{-3} \). We know that \( e^{-3}=\frac{1}{e^{3}}\approx\frac{1}{20.0855}\approx0.0498 \). Then multiply by 5: \( 5\times0.0498 = 0.249 \), and add 1: \( 0.249 + 1=1.249 \).

Step2: For \( x = -2 \)

Substitute \( x=-2 \) into \( f(x) \). \( e^{-2}=\frac{1}{e^{2}}\approx\frac{1}{7.3891}\approx0.1353 \). Multiply by 5: \( 5\times0.1353 = 0.6765 \), add 1: \( 0.6765+1 = 1.677 \) (rounded to three decimal places).

Step3: For \( x=-1 \)

Substitute \( x = -1 \) into \( f(x) \). \( e^{-1}=\frac{1}{e}\approx0.3679 \). Multiply by 5: \( 5\times0.3679=1.8395 \), add 1: \( 1.8395 + 1=2.840 \) (rounded to three decimal places).

Step4: For \( x = 0 \)

Substitute \( x = 0 \) into \( f(x) \). \( e^{0}=1 \). Then \( 5\times1+1=5 + 1=6.000 \).

Step5: For \( x = 1 \)

Substitute \( x = 1 \) into \( f(x) \). \( e^{1}\approx2.7183 \). Multiply by 5: \( 5\times2.7183 = 13.5915 \), add 1: \( 13.5915+1=14.592 \) (rounded to three decimal places).

Answer:

\( x \)	\( -3 \)	\( -2 \)	\( -1 \)	\( 0 \)	\( 1 \)

For the graph sketching, the function \( f(x)=5e^{x}+1 \) is an exponential growth function (since the exponent is \( x \), positive coefficient of \( x \) in the exponent) with vertical shift up by 1. As \( x\to-\infty \), \( e^{x}\to0 \), so \( f(x)\to1 \), and as \( x\to\infty \), \( f(x)\to\infty \). So the correct graph should be the one that approaches \( y = 1 \) as \( x\to-\infty \) and increases rapidly as \( x \) increases, which is the third graph (the one with the curve starting near \( y = 1 \) on the left and rising on the right, similar to the standard exponential growth shifted up). But since the question mainly asked for the table, the table values are as above.