Question

complete the table of values for the functions $f(x) = 4|x|$ and $g(x) = -2(x - 1)^2 + 12$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \

$$\begin{tabular}{|c|c|c|} \\hline $x$ & $f(x)$ & $g(x)$ \\\\ \\hline $-3$ & $12$ & $-20$ \\\\ \\hline $-2$ & $\\square$ & $\\square$ \\\\ \\hline $-1$ & $\\square$ & $\\square$ \\\\ \\hline $0$ & $0$ & $10$ \\\\ \\hline $1$ & $\\square$ & $\\square$ \\\\ \\hline \\end{tabular}$$

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution? $x = -2$ \quad\quad\quad\quad between $x = -2$ and $x = -1$ $x = -1$ \quad\quad\quad\quad between $x = -1$ and $x = 0$

Explanation:

Part 1: Completing the table for \( f(x) = 4|x| \) and \( g(x) = -2(x - 1)^2 + 12 \)

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

• When \( x = -2 \):

\( f(-2) = 4|-2| = 4 \times 2 = 8 \)

• When \( x = -1 \):

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

• When \( x = 1 \):

\( f(1) = 4|1| = 4 \times 1 = 4 \)

For \( g(x) = -2(x - 1)^2 + 12 \):

• When \( x = -2 \):

\( g(-2) = -2(-2 - 1)^2 + 12 = -2(-3)^2 + 12 = -2(9) + 12 = -18 + 12 = -6 \)

• When \( x = -1 \):

\( g(-1) = -2(-1 - 1)^2 + 12 = -2(-2)^2 + 12 = -2(4) + 12 = -8 + 12 = 4 \)

• When \( x = 1 \):

\( g(1) = -2(1 - 1)^2 + 12 = -2(0)^2 + 12 = 0 + 12 = 12 \)

Filled Table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -2 \)	\( 8 \)	\( -6 \)
\( -1 \)	\( 4 \)	\( 4 \)
\( 0 \)	\( 0 \)	\( 10 \)
\( 1 \)	\( 4 \)	\( 12 \)

Part 2: Solving \( f(x) = g(x) \)

To find where \( f(x) = g(x) \), we analyze the table:

• At \( x = -2 \): \( f(-2) = 8 \), \( g(-2) = -6 \) ( \( f(x) > g(x) \) )
• At \( x = -1 \): \( f(-1) = 4 \), \( g(-1) = 4 \) ( \( f(x) = g(x) \) ) Wait, no—wait, the table shows \( f(-1) = 4 \) and \( g(-1) = 4 \), so \( x = -1 \) is a solution? Wait, no, let’s recheck:

Wait, the original table for \( x = -1 \):
\( f(-1) = 4 \), \( g(-1) = 4 \). So \( f(-1) = g(-1) \), meaning \( x = -1 \) is a solution? But the options include "between \( x = -2 \) and \( x = -1 \)" or "between \( x = -1 \) and \( x = 0 \)". Wait, maybe a miscalculation?

Wait, let’s re-express \( g(x) \):
\( g(x) = -2(x - 1)^2 + 12 \). At \( x = -1 \):
\( (x - 1) = -2 \), squared is \( 4 \), times \( -2 \) is \( -8 \), plus \( 12 \) is \( 4 \). Correct. \( f(-1) = 4 \). So \( f(-1) = g(-1) \), so \( x = -1 \) is a solution. But the options are:

• \( x = -2 \)
• between \( x = -2 \) and \( x = -1 \)
• \( x = -1 \)
• between \( x = -1 \) and \( x = 0 \)

From the table, at \( x = -1 \), \( f(x) = g(x) = 4 \). So the solution is \( x = -1 \).

Final Answers:

Table Completion:

• \( x = -2 \): \( f(x) = 8 \), \( g(x) = -6 \)
• \( x = -1 \): \( f(x) = 4 \), \( g(x) = 4 \)
• \( x = 1 \): \( f(x) = 4 \), \( g(x) = 12 \)

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

\( x = -1 \) (since \( f(-1) = g(-1) = 4 \))

Final Answer for the Equation:

\( \boldsymbol{x = -1} \)

Answer:

Part 1: Completing the table for \( f(x) = 4|x| \) and \( g(x) = -2(x - 1)^2 + 12 \)

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

• When \( x = -2 \):

\( f(-2) = 4|-2| = 4 \times 2 = 8 \)

• When \( x = -1 \):

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

• When \( x = 1 \):

\( f(1) = 4|1| = 4 \times 1 = 4 \)

For \( g(x) = -2(x - 1)^2 + 12 \):

• When \( x = -2 \):

\( g(-2) = -2(-2 - 1)^2 + 12 = -2(-3)^2 + 12 = -2(9) + 12 = -18 + 12 = -6 \)

• When \( x = -1 \):

\( g(-1) = -2(-1 - 1)^2 + 12 = -2(-2)^2 + 12 = -2(4) + 12 = -8 + 12 = 4 \)

• When \( x = 1 \):

\( g(1) = -2(1 - 1)^2 + 12 = -2(0)^2 + 12 = 0 + 12 = 12 \)

Filled Table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -2 \)	\( 8 \)	\( -6 \)
\( -1 \)	\( 4 \)	\( 4 \)
\( 0 \)	\( 0 \)	\( 10 \)
\( 1 \)	\( 4 \)	\( 12 \)

Part 2: Solving \( f(x) = g(x) \)

To find where \( f(x) = g(x) \), we analyze the table:

• At \( x = -2 \): \( f(-2) = 8 \), \( g(-2) = -6 \) ( \( f(x) > g(x) \) )
• At \( x = -1 \): \( f(-1) = 4 \), \( g(-1) = 4 \) ( \( f(x) = g(x) \) ) Wait, no—wait, the table shows \( f(-1) = 4 \) and \( g(-1) = 4 \), so \( x = -1 \) is a solution? Wait, no, let’s recheck:

Wait, the original table for \( x = -1 \):
\( f(-1) = 4 \), \( g(-1) = 4 \). So \( f(-1) = g(-1) \), meaning \( x = -1 \) is a solution? But the options include "between \( x = -2 \) and \( x = -1 \)" or "between \( x = -1 \) and \( x = 0 \)". Wait, maybe a miscalculation?

Wait, let’s re-express \( g(x) \):
\( g(x) = -2(x - 1)^2 + 12 \). At \( x = -1 \):
\( (x - 1) = -2 \), squared is \( 4 \), times \( -2 \) is \( -8 \), plus \( 12 \) is \( 4 \). Correct. \( f(-1) = 4 \). So \( f(-1) = g(-1) \), so \( x = -1 \) is a solution. But the options are:

• \( x = -2 \)
• between \( x = -2 \) and \( x = -1 \)
• \( x = -1 \)
• between \( x = -1 \) and \( x = 0 \)

From the table, at \( x = -1 \), \( f(x) = g(x) = 4 \). So the solution is \( x = -1 \).

Final Answers:

Table Completion:

• \( x = -2 \): \( f(x) = 8 \), \( g(x) = -6 \)
• \( x = -1 \): \( f(x) = 4 \), \( g(x) = 4 \)
• \( x = 1 \): \( f(x) = 4 \), \( g(x) = 12 \)

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

\( x = -1 \) (since \( f(-1) = g(-1) = 4 \))

Final Answer for the Equation:

\( \boldsymbol{x = -1} \)