Question

complete the table of values for the functions $f(x) = -3x + 7$ and $g(x) = 2^x + 1$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \

$$\begin{tabular}{|c|c|c|} \\hline \\textbf{$x$} & \\textbf{$f(x)$} & \\textbf{$g(x)$} \\\\ \\hline 0 & 7 & 2 \\\\ \\hline 1 & \\fbox{} & \\fbox{} \\\\ \\hline 2 & \\fbox{} & \\fbox{} \\\\ \\hline 3 & -2 & 9 \\\\ \\hline 4 & \\fbox{} & \\fbox{} \\\\ \\hline \\end{tabular}$$

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution? $x = 1$; between $x = 1$ and $x = 2$; $x = 2$; between $x = 2$ and $x = 3$

Explanation:

Part 1: Completing the table for \( f(x) = -3x + 7 \) and \( g(x) = 2^x + 1 \)

For \( x = 1 \):

• Step 1: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) = -3x + 7 \):
\( f(1) = -3(1) + 7 = -3 + 7 = 4 \)

• Step 2: Calculate \( g(1) \)

Substitute \( x = 1 \) into \( g(x) = 2^x + 1 \):
\( g(1) = 2^1 + 1 = 2 + 1 = 3 \)

For \( x = 2 \):

• Step 1: Calculate \( f(2) \)

Substitute \( x = 2 \) into \( f(x) = -3x + 7 \):
\( f(2) = -3(2) + 7 = -6 + 7 = 1 \)

• Step 2: Calculate \( g(2) \)

Substitute \( x = 2 \) into \( g(x) = 2^x + 1 \):
\( g(2) = 2^2 + 1 = 4 + 1 = 5 \)

For \( x = 4 \):

• Step 1: Calculate \( f(4) \)

Substitute \( x = 4 \) into \( f(x) = -3x + 7 \):
\( f(4) = -3(4) + 7 = -12 + 7 = -5 \)

• Step 2: Calculate \( g(4) \)

Substitute \( x = 4 \) into \( g(x) = 2^x + 1 \):
\( g(4) = 2^4 + 1 = 16 + 1 = 17 \)

Completed Table:

\( x \)	\( f(x) \)	\( g(x) \)
1	4	3
2	1	5
3	-2	9
4	-5	17

Part 2: Finding where \( f(x) = g(x) \)

To determine where \( f(x) = g(x) \), we analyze the values of \( f(x) \) and \( g(x) \) at each \( x \):

• At \( x = 1 \): \( f(1) = 4 \), \( g(1) = 3 \) → \( f(x) > g(x) \)
• At \( x = 2 \): \( f(2) = 1 \), \( g(2) = 5 \) → \( f(x) < g(x) \)

Since \( f(x) \) is linear (decreasing) and \( g(x) \) is exponential (increasing), the function \( f(x) \) crosses \( g(x) \) between \( x = 1 \) and \( x = 2 \) (because \( f(x) \) goes from above \( g(x) \) to below \( g(x) \) in this interval).

Final Answers:

• Table values:

\( x = 1 \): \( f(1) = 4 \), \( g(1) = 3 \)
\( x = 2 \): \( f(2) = 1 \), \( g(2) = 5 \)
\( x = 4 \): \( f(4) = -5 \), \( g(4) = 17 \)

• Solution to \( f(x) = g(x) \): between \( x = 1 \) and \( x = 2 \)

Table (Final):

\( x \)	\( f(x) \)	\( g(x) \)
1	4	3
2	1	5
3	-2	9
4	-5	17

Answer:

for \( f(x) = g(x) \):
between \( x = 1 \) and \( x = 2 \)