Question

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

$$\begin{tabular}{|c|c|c|} \\hline \\( x \\) & \\( f(x) \\) & \\( g(x) \\) \\\\ \\hline 3 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 4 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 5 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 6 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline 7 & \\( \\square \\) & \\( \\square \\) \\\\ \\hline \\end{tabular}$$

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution? between $x = 4$ and $x = 5$ \quad \quad \quad \quad $x = 5$ between $x = 5$ and $x = 6$ \quad \quad \quad \quad $x = 6$

Explanation:

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

For \( f(x) = |x - 7| \):

• When \( x = 3 \):
• Step 1: Substitute \( x = 3 \) into \( f(x) \).
• \( f(3) = |3 - 7| = |-4| = 4 \)
• When \( x = 4 \):
• Step 1: Substitute \( x = 4 \) into \( f(x) \).
• \( f(4) = |4 - 7| = |-3| = 3 \)
• When \( x = 5 \):
• Step 1: Substitute \( x = 5 \) into \( f(x) \).
• \( f(5) = |5 - 7| = |-2| = 2 \)
• When \( x = 6 \):
• Step 1: Substitute \( x = 6 \) into \( f(x) \).
• \( f(6) = |6 - 7| = |-1| = 1 \)
• When \( x = 7 \):
• Step 1: Substitute \( x = 7 \) into \( f(x) \).
• \( f(7) = |7 - 7| = |0| = 0 \)

For \( g(x) = (x - 3)^2 \):

• When \( x = 3 \):
• Step 1: Substitute \( x = 3 \) into \( g(x) \).
• \( g(3) = (3 - 3)^2 = 0^2 = 0 \)
• When \( x = 4 \):
• Step 1: Substitute \( x = 4 \) into \( g(x) \).
• \( g(4) = (4 - 3)^2 = 1^2 = 1 \)
• When \( x = 5 \):
• Step 1: Substitute \( x = 5 \) into \( g(x) \).
• \( g(5) = (5 - 3)^2 = 2^2 = 4 \)
• When \( x = 6 \):
• Step 1: Substitute \( x = 6 \) into \( g(x) \).
• \( g(6) = (6 - 3)^2 = 3^2 = 9 \)
• When \( x = 7 \):
• Step 1: Substitute \( x = 7 \) into \( g(x) \).
• \( g(7) = (7 - 3)^2 = 4^2 = 16 \)

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

We analyze the values of \( f(x) \) and \( g(x) \) at each \( x \):

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

Since \( f(x) \) decreases from 3 to 2 and \( g(x) \) increases from 1 to 4 between \( x = 4 \) and \( x = 5 \), the equation \( f(x) = g(x) \) must have a solution between \( x = 4 \) and \( x = 5 \) (by the Intermediate Value Theorem, as the functions are continuous).

Final Table:

\( x \)	\( f(x) \)	\( g(x) \)
4	3	1
5	2	4
6	1	9
7	0	16

Answer:

for the solution of \( f(x) = g(x) \):
The equation \( f(x) = g(x) \) has a solution between \( x = 4 \) and \( x = 5 \).