Question

complete the table of values for the functions $f(x)=|x + 4|$ and $g(x)=x^{2}-1$. write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. \
(table with x, f(x), g(x) rows: -2 (f(x) blank, g(x) blank), -1 (f(x)=3, g(x)=0), 0 (f(x) blank, g(x) blank), 1 (f(x) blank, g(x) blank), 2 (f(x)=6, g(x)=3)) \
based on the values in the table, where does the equation $f(x)=g(x)$ have a solution? \
$x = -2$ between $x = -2$ and $x = -1$ \
$x = -1$ between $x = -1$ and $x = 0$

Explanation:

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

For \( x = -2 \):

• Step 1: Calculate \( f(-2) \)

Substitute \( x = -2 \) into \( f(x) = |x + 4| \):
\( f(-2) = |-2 + 4| = |2| = 2 \)

• Step 2: Calculate \( g(-2) \)

Substitute \( x = -2 \) into \( g(x) = x^2 - 1 \):
\( g(-2) = (-2)^2 - 1 = 4 - 1 = 3 \)

For \( x = 0 \):

• Step 1: Calculate \( f(0) \)

Substitute \( x = 0 \) into \( f(x) = |x + 4| \):
\( f(0) = |0 + 4| = |4| = 4 \)

• Step 2: Calculate \( g(0) \)

Substitute \( x = 0 \) into \( g(x) = x^2 - 1 \):
\( g(0) = 0^2 - 1 = 0 - 1 = -1 \)

For \( x = 1 \):

• Step 1: Calculate \( f(1) \)

Substitute \( x = 1 \) into \( f(x) = |x + 4| \):
\( f(1) = |1 + 4| = |5| = 5 \)

• Step 2: Calculate \( g(1) \)

Substitute \( x = 1 \) into \( g(x) = x^2 - 1 \):
\( g(1) = 1^2 - 1 = 1 - 1 = 0 \)

Part 2: Determine where \( f(x) = g(x) \) has a solution

We analyze the table values:

• At \( x = -2 \): \( f(-2) = 2 \), \( g(-2) = 3 \) (so \( f(x) < g(x) \))
• At \( x = -1 \): \( f(-1) = 3 \), \( g(-1) = 0 \) (so \( f(x) > g(x) \))

Since \( f(x) \) and \( g(x) \) are continuous (absolute value and quadratic functions are continuous), by the Intermediate Value Theorem, if \( f(x) \) changes from less than \( g(x) \) to greater than \( g(x) \) between \( x = -2 \) and \( x = -1 \), there must be a solution to \( f(x) = g(x) \) in this interval.

Completed Table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -1 \)	\( 3 \)	\( 0 \)
\( 0 \)	\( 4 \)	\( -1 \)
\( 1 \)	\( 5 \)	\( 0 \)
\( 2 \)	\( 6 \)	\( 3 \)

Answer:

to \( f(x) = g(x) \) solution:
The equation \( f(x) = g(x) \) has a solution between \( x = -2 \) and \( x = -1 \).