Question

look at the table of values for the functions $f(x) = |x| + 1.5$ and $g(x) = 0.5x^2$.

$x$	$f(x)$	$g(x)$
1	2.5	0.5
2	3.5	2
3	4.5	4.5
4	5.5	8

based on the values in the table, where does the equation $f(x) = g(x)$ have a solution?
between $x = 2$ and $x = 3$ $x = 3$
between $x = 3$ and $x = 4$ $x = 4$

Explanation:

Step1: Check x=3 values

For \( x = 3 \), \( f(3)=|3| + 1.5=4.5 \) and \( g(3)=0.5\times3^{2}=4.5 \). So \( f(3)=g(3) \).

Step2: Verify other options

• Between \( x = 2 \) and \( x = 3 \): At \( x = 2 \), \( f(2)=3.5 \), \( g(2)=2 \) (\( f(2)>g(2) \)); at \( x = 3 \), \( f(3)=g(3) \), but the solution is exactly at \( x = 3 \), not between 2 and 3.
• Between \( x = 3 \) and \( x = 4 \): At \( x = 3 \), \( f(3)=g(3) \); at \( x = 4 \), \( f(4)=5.5 \), \( g(4)=8 \) (\( f(4)<g(4) \)), but the solution is at \( x = 3 \), not between 3 and 4.
• \( x = 4 \): \( f(4)=5.5 \), \( g(4)=8 \), not equal.

Answer:

\( x = 3 \)