Question

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

$x$	$f(x)$	$g(x)$
$-1$	$\square$	$\square$
$0$	$\square$	$\square$
$1$	$\square$	$\square$
$2$	$\square$	$\square$

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

Explanation:

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

For \( f(x) = \frac{1}{2}|x| \):

• When \( x = -2 \):

\( f(-2) = \frac{1}{2}|-2| = \frac{1}{2} \times 2 = 1 \)

• When \( x = -1 \):

\( f(-1) = \frac{1}{2}|-1| = \frac{1}{2} \times 1 = 0.5 \)

• When \( x = 0 \):

\( f(0) = \frac{1}{2}|0| = \frac{1}{2} \times 0 = 0 \)

• When \( x = 1 \):

\( f(1) = \frac{1}{2}|1| = \frac{1}{2} \times 1 = 0.5 \)

• When \( x = 2 \):

\( f(2) = \frac{1}{2}|2| = \frac{1}{2} \times 2 = 1 \)

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

• When \( x = -2 \):

\( g(-2) = 2^{-2} = \frac{1}{2^2} = \frac{1}{4} = 0.25 \)

• When \( x = -1 \):

\( g(-1) = 2^{-1} = \frac{1}{2} = 0.5 \)

• When \( x = 0 \):

\( g(0) = 2^0 = 1 \)

• When \( x = 1 \):

\( g(1) = 2^1 = 2 \)

• When \( x = 2 \):

\( g(2) = 2^2 = 4 \)

Filled Table:

\( x \)	\( f(x) \)	\( g(x) \)
\( -1 \)	\( 0.5 \)	\( 0.5 \)
\( 0 \)	\( 0 \)	\( 1 \)
\( 1 \)	\( 0.5 \)	\( 2 \)
\( 2 \)	\( 1 \)	\( 4 \)

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

We analyze the table values:

• At \( x = -2 \): \( f(-2) = 1 \), \( g(-2) = 0.25 \) ( \( f > g \) )
• At \( x = -1 \): \( f(-1) = 0.5 \), \( g(-1) = 0.5 \) ( \( f = g \) )

Answer:

(Table Completion):

\( x \)	\( f(x) \)	\( g(x) \)
\( -1 \)	\( 0.5 \)	\( 0.5 \)
\( 0 \)	\( 0 \)	\( 1 \)
\( 1 \)	\( 0.5 \)	\( 2 \)
\( 2 \)	\( 1 \)	\( 4 \)