Question

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

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

based on the values in the table, where does the equation ( f(x) = g(x) ) have a solution?

between ( x = -3 ) and ( x = -2 ) ( x = -2 )
between ( x = -2 ) and ( x = -1 ) ( x = -1 )

Explanation:

Part 1: Completing the table for \( f(x) = |2x| \) and \( g(x) = 3^{x + 2} \)

For \( f(x) = |2x| \):

• Step 1: When \( x = -4 \)

Substitute \( x = -4 \) into \( f(x) \):
\( f(-4) = |2(-4)| = |-8| = 8 \)

• Step 2: When \( x = -3 \)

Substitute \( x = -3 \):
\( f(-3) = |2(-3)| = |-6| = 6 \)

• Step 3: When \( x = -2 \)

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

• Step 4: When \( x = -1 \)

Substitute \( x = -1 \):
\( f(-1) = |2(-1)| = |-2| = 2 \)

• Step 5: When \( x = 0 \)

Substitute \( x = 0 \):
\( f(0) = |2(0)| = |0| = 0 \)

For \( g(x) = 3^{x + 2} \):

• Step 1: When \( x = -4 \)

Substitute \( x = -4 \):
\( g(-4) = 3^{-4 + 2} = 3^{-2} = \frac{1}{3^2} = \frac{1}{9} \approx 0.111 \)

• Step 2: When \( x = -3 \)

Substitute \( x = -3 \):
\( g(-3) = 3^{-3 + 2} = 3^{-1} = \frac{1}{3} \approx 0.333 \)

• Step 3: When \( x = -2 \)

Substitute \( x = -2 \):
\( g(-2) = 3^{-2 + 2} = 3^{0} = 1 \)

• Step 4: When \( x = -1 \)

Substitute \( x = -1 \):
\( g(-1) = 3^{-1 + 2} = 3^{1} = 3 \)

• Step 5: When \( x = 0 \)

Substitute \( x = 0 \):
\( g(0) = 3^{0 + 2} = 3^{2} = 9 \)

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

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

\( x \)	\( f(x) \)	\( g(x) \)	Relationship
\( -3 \)	\( 6 \)	\( \frac{1}{3} \approx 0.333 \)	\( f(x) > g(x) \)
\( -2 \)	\( 4 \)	\( 1 \)	\( f(x) > g(x) \)
\( -1 \)	\( 2 \)	\( 3 \)	\( f(x) < g(x) \)

At \( x = -2 \), \( f(-2) = 4 \) and \( g(-2) = 1 \) (so \( f > g \)).
At \( x = -1 \), \( f(-1) = 2 \) and \( g(-1) = 3 \) (so \( f < g \)).

Since \( f(x) \) is decreasing (or constant) and \( g(x) \) is increasing, the solution to \( f(x) = g(x) \) lies between \( x = -2 \) and \( x = -1 \) (because \( f(x) \) crosses from above \( g(x) \) to below \( g(x) \) in this interval).

Final Table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -3 \)	\( 6 \)	\( \frac{1}{3} \) (or \( 0.333 \))
\( -2 \)	\( 4 \)	\( 1 \)
\( -1 \)	\( 2 \)	\( 3 \)
\( 0 \)	\( 0 \)	\( 9 \)

Answer:

to \( f(x) = g(x) \):
between \( x = -2 \) and \( x = -1 \)