Question

complete the table of values for the functions $f(x)=|x - 3|-4$ and $g(x)=3^x - 5$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \

$$\begin{tabular}{|c|c|c|} \\hline \\( x \\) & \\( f(x) \\) & \\( g(x) \\) \\\\ \\hline 0 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 1 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 2 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 3 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 4 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline \\end{tabular}$$

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 \( f(x) = |x - 3| - 4 \) and \( g(x) = 3^x - 5 \)

For \( f(x) = |x - 3| - 4 \):

• When \( x = 0 \):

Step1: Substitute \( x = 0 \)

\( f(0) = |0 - 3| - 4 = |-3| - 4 = 3 - 4 = -1 \)

• When \( x = 1 \):

Step1: Substitute \( x = 1 \)

\( f(1) = |1 - 3| - 4 = |-2| - 4 = 2 - 4 = -2 \)

• When \( x = 2 \):

Step1: Substitute \( x = 2 \)

\( f(2) = |2 - 3| - 4 = |-1| - 4 = 1 - 4 = -3 \)

• When \( x = 3 \):

Step1: Substitute \( x = 3 \)

\( f(3) = |3 - 3| - 4 = |0| - 4 = 0 - 4 = -4 \)

• When \( x = 4 \):

Step1: Substitute \( x = 4 \)

\( f(4) = |4 - 3| - 4 = |1| - 4 = 1 - 4 = -3 \)

For \( g(x) = 3^x - 5 \):

• When \( x = 0 \):

Step1: Substitute \( x = 0 \)

\( g(0) = 3^0 - 5 = 1 - 5 = -4 \)

• When \( x = 1 \):

Step1: Substitute \( x = 1 \)

\( g(1) = 3^1 - 5 = 3 - 5 = -2 \)

• When \( x = 2 \):

Step1: Substitute \( x = 2 \)

\( g(2) = 3^2 - 5 = 9 - 5 = 4 \)

• When \( x = 3 \):

Step1: Substitute \( x = 3 \)

\( g(3) = 3^3 - 5 = 27 - 5 = 22 \)

• When \( x = 4 \):

Step1: Substitute \( x = 4 \)

\( g(4) = 3^4 - 5 = 81 - 5 = 76 \)

Filled Table:

\( x \)	\( f(x) \)	\( g(x) \)
1	-2	-2
2	-3	4
3	-4	22
4	-3	76

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

We analyze the values:

• At \( x = 1 \): \( f(1) = -2 \) and \( g(1) = -2 \). Wait, but let's check again. Wait, at \( x = 1 \), \( f(1) = -2 \) and \( g(1) = -2 \), so \( f(1) = g(1) \)? Wait, no, wait the table:

Wait, when \( x = 1 \):

• \( f(1) = |1 - 3| - 4 = 2 - 4 = -2 \)
• \( g(1) = 3^1 - 5 = 3 - 5 = -2 \)

Wait, but the options include "between \( x = 1 \) and \( x = 2 \)", "between \( x = 2 \) and \( x = 3 \)", etc. Wait, maybe I made a mistake. Wait, no, at \( x = 1 \), \( f(x) = g(x) = -2 \), so \( x = 1 \) is a solution? But let's check the options. The options are:

• \( x = 1 \)
• between \( x = 1 \) and \( x = 2 \)
• \( x = 2 \)
• between \( x = 2 \) and \( x = 3 \)

Wait, at \( x = 1 \), \( f(1) = -2 \) and \( g(1) = -2 \), so \( f(1) = g(1) \). So the solution is \( x = 1 \). But wait, let's confirm:

Wait, \( f(1) = |1 - 3| - 4 = 2 - 4 = -2 \)
\( g(1) = 3^1 - 5 = 3 - 5 = -2 \)

So \( f(1) = g(1) \), so \( x = 1 \) is a solution.

But wait, the options include "between \( x = 1 \) and \( x = 2 \)", but at \( x = 1 \), they are equal. So the answer is \( x = 1 \).

Final Answers:

Table:

\( x \)	\( f(x) \)	\( g(x) \)
1	-2	-2
2	-3	4
3	-4	22
4	-3	76

Solution to \( f(x) = g(x) \):

\( x = 1 \)

Answer:

(Table):

\( x \)	\( f(x) \)	\( g(x) \)
1	-2	-2
2	-3	4
3	-4	22
4	-3	76