Question

look at the table of values for the functions $f(x) = |x| - 3$ and $g(x) = 3^{x + 4}$.

$x$	$f(x)$	$g(x)$
$-6$	$3$	$\frac{1}{9}$
$-5$	$2$	$\frac{1}{3}$
$-4$	$1$	$1$
$-3$	$0$	$3$

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
between $x = -5$ and $x = -4$
$x = -4$
between $x = -4$ and $x = -3$
$x = -3$

Explanation:

Step1: Check x = -4 values

At \( x = -4 \), \( f(-4)=| - 4|-3 = 1 \) and \( g(-4)=3^{-4 + 4}=3^{0}=1 \). So \( f(-4)=g(-4) \).

Step2: Verify other intervals

• For \( x \) between -5 and -4: At \( x=-5 \), \( f(-5)=2 \), \( g(-5)=\frac{1}{3}\) (\( f(-5)>g(-5) \)); at \( x = - 4 \), \( f(-4)=g(-4) \). But since at \( x=-4 \) they are equal, the solution is at \( x=-4 \), not between -5 and -4.
• For \( x \) between -4 and -3: At \( x=-4 \), \( f(-4)=g(-4) = 1 \); at \( x=-3 \), \( f(-3)=0 \), \( g(-3)=3 \) (\( f(-3)<g(-3) \)), so no solution here.
• At \( x=-3 \), \( f(-3)=0

eq g(-3)=3 \).

Answer:

\( x = -4 \)