Question

complete the table of values for the functions $f(x) = 2|x|$ and $g(x) = x^2 - 1$.
write your answers as whole numbers, decimals, or simplified fractions or mixed numbers.
(table with columns x, f(x), g(x) and rows -3, -2, -1, 0, 1, with empty boxes for f(x) and g(x) values)
based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
options: between $x = -3$ and $x = -2$; $x = -2$; between $x = -2$ and $x = -1$; $x = -1$

Explanation:

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

For \( x = -3 \):

• Calculating \( f(-3) \):

The function is \( f(x) = 2|x| \). Substitute \( x = -3 \):
\( |-3| = 3 \), so \( f(-3) = 2 \times 3 = 6 \).

• Calculating \( g(-3) \):

The function is \( g(x) = x^2 - 1 \). Substitute \( x = -3 \):
\( (-3)^2 = 9 \), so \( g(-3) = 9 - 1 = 8 \).

For \( x = -2 \):

• Calculating \( f(-2) \):

\( |-2| = 2 \), so \( f(-2) = 2 \times 2 = 4 \).

• Calculating \( g(-2) \):

\( (-2)^2 = 4 \), so \( g(-2) = 4 - 1 = 3 \).

For \( x = -1 \):

• Calculating \( f(-1) \):

\( |-1| = 1 \), so \( f(-1) = 2 \times 1 = 2 \).

• Calculating \( g(-1) \):

\( (-1)^2 = 1 \), so \( g(-1) = 1 - 1 = 0 \).

For \( x = 0 \):

• Calculating \( f(0) \):

\( |0| = 0 \), so \( f(0) = 2 \times 0 = 0 \).

• Calculating \( g(0) \):

\( 0^2 = 0 \), so \( g(0) = 0 - 1 = -1 \).

For \( x = 1 \):

• Calculating \( f(1) \):

\( |1| = 1 \), so \( f(1) = 2 \times 1 = 2 \).

• Calculating \( g(1) \):

\( 1^2 = 1 \), so \( g(1) = 1 - 1 = 0 \).

The completed table is:

\( x \)	\( f(x) \)	\( g(x) \)
\( -2 \)	\( 4 \)	\( 3 \)
\( -1 \)	\( 2 \)	\( 0 \)
\( 0 \)	\( 0 \)	\( -1 \)
\( 1 \)	\( 2 \)	\( 0 \)

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 \( x = -3 \): \( f(-3) = 6 \), \( g(-3) = 8 \) (so \( f(x) < g(x) \)).
• At \( x = -2 \): \( f(-2) = 4 \), \( g(-2) = 3 \) (so \( f(x) > g(x) \)).

Since \( f(x) \) changes from being less than \( g(x) \) (at \( x = -3 \)) to greater than \( g(x) \) (at \( x = -2 \)), by the Intermediate Value Theorem, the equation \( f(x) = g(x) \) must have a solution between \( x = -3 \) and \( x = -2 \).

Final Answers

Table Completion:

\( x \)	\( f(x) \)	\( g(x) \)
\( -2 \)	\( 4 \)	\( 3 \)
\( -1 \)	\( 2 \)	\( 0 \)
\( 0 \)	\( 0 \)	\( -1 \)
\( 1 \)	\( 2 \)	\( 0 \)

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

The equation \( f(x) = g(x) \) has a solution between \( x = -3 \) and \( x = -2 \).

Answer:

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

For \( x = -3 \):

• Calculating \( f(-3) \):

The function is \( f(x) = 2|x| \). Substitute \( x = -3 \):
\( |-3| = 3 \), so \( f(-3) = 2 \times 3 = 6 \).

• Calculating \( g(-3) \):

The function is \( g(x) = x^2 - 1 \). Substitute \( x = -3 \):
\( (-3)^2 = 9 \), so \( g(-3) = 9 - 1 = 8 \).

For \( x = -2 \):

• Calculating \( f(-2) \):

\( |-2| = 2 \), so \( f(-2) = 2 \times 2 = 4 \).

• Calculating \( g(-2) \):

\( (-2)^2 = 4 \), so \( g(-2) = 4 - 1 = 3 \).

For \( x = -1 \):

• Calculating \( f(-1) \):

\( |-1| = 1 \), so \( f(-1) = 2 \times 1 = 2 \).

• Calculating \( g(-1) \):

\( (-1)^2 = 1 \), so \( g(-1) = 1 - 1 = 0 \).

For \( x = 0 \):

• Calculating \( f(0) \):

\( |0| = 0 \), so \( f(0) = 2 \times 0 = 0 \).

• Calculating \( g(0) \):

\( 0^2 = 0 \), so \( g(0) = 0 - 1 = -1 \).

For \( x = 1 \):

• Calculating \( f(1) \):

\( |1| = 1 \), so \( f(1) = 2 \times 1 = 2 \).

• Calculating \( g(1) \):

\( 1^2 = 1 \), so \( g(1) = 1 - 1 = 0 \).

The completed table is:

\( x \)	\( f(x) \)	\( g(x) \)
\( -2 \)	\( 4 \)	\( 3 \)
\( -1 \)	\( 2 \)	\( 0 \)
\( 0 \)	\( 0 \)	\( -1 \)
\( 1 \)	\( 2 \)	\( 0 \)

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 \( x = -3 \): \( f(-3) = 6 \), \( g(-3) = 8 \) (so \( f(x) < g(x) \)).
• At \( x = -2 \): \( f(-2) = 4 \), \( g(-2) = 3 \) (so \( f(x) > g(x) \)).

Since \( f(x) \) changes from being less than \( g(x) \) (at \( x = -3 \)) to greater than \( g(x) \) (at \( x = -2 \)), by the Intermediate Value Theorem, the equation \( f(x) = g(x) \) must have a solution between \( x = -3 \) and \( x = -2 \).

Final Answers

Table Completion:

\( x \)	\( f(x) \)	\( g(x) \)
\( -2 \)	\( 4 \)	\( 3 \)
\( -1 \)	\( 2 \)	\( 0 \)
\( 0 \)	\( 0 \)	\( -1 \)
\( 1 \)	\( 2 \)	\( 0 \)

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

The equation \( f(x) = g(x) \) has a solution between \( x = -3 \) and \( x = -2 \).