Question

sara wants to find the input value that produces the same output for the functions represented by the tables.

$f(x) = -0.5x + 2$			$g(x) = 2x - 3$
---------	---------		---------	---------
$x$	$f(x)$		$x$	$g(x)$
$-3$	$3.5$		$-3$
$-2$	$3$		$-2$
$-1$	$2.5$		$-1$
$0$	$2$		$0$
$1$	$1.5$		$1$
$2$	$1$		$2$
$3$	$0.5$		$3$

what is the input value that produces the same output value in both charts?

• $-2$
• $-1$
• $1$
• $2$

Explanation:

Step1: Set the functions equal

To find the input value \( x \) that produces the same output for \( f(x) \) and \( g(x) \), we set \( f(x) = g(x) \). So, \(-0.5x + 2 = 2x - 3\).

Step2: Solve for \( x \)

First, add \( 0.5x \) to both sides: \( 2 = 2x + 0.5x - 3 \).
Simplify the right side: \( 2 = 2.5x - 3 \).
Then, add 3 to both sides: \( 2 + 3 = 2.5x \), which gives \( 5 = 2.5x \).
Divide both sides by \( 2.5 \): \( x = \frac{5}{2.5} = 2 \). Wait, but let's check with the tables. Wait, maybe I made a mistake. Wait, let's check the tables. For \( f(x) = -0.5x + 2 \) and \( g(x) = 2x - 3 \), let's plug in \( x = 2 \): \( f(2) = -0.5(2) + 2 = -1 + 2 = 1 \). \( g(2) = 2(2) - 3 = 4 - 3 = 1 \). Wait, but let's check the options. Wait, the options are -2, -1, 1, 2. Wait, when \( x = 2 \), both outputs are 1. Wait, but let's check \( x = 2 \) in the tables. For \( f(x) \), when \( x = 2 \), \( f(x) = 1 \). For \( g(x) \), when \( x = 2 \), \( g(x) = 1 \). So that's correct. Wait, but let's check the equation again. \(-0.5x + 2 = 2x - 3\). Let's solve again:
\(-0.5x - 2x = -3 - 2\)
\(-2.5x = -5\)
\(x = \frac{-5}{-2.5} = 2\). Yes, so \( x = 2 \). But wait, let's check the other options. For \( x = 1 \): \( f(1) = -0.5(1) + 2 = 1.5 \), \( g(1) = 2(1) - 3 = -1 \). Not equal. For \( x = -1 \): \( f(-1) = -0.5(-1) + 2 = 0.5 + 2 = 2.5 \), \( g(-1) = 2(-1) - 3 = -2 - 3 = -5 \). Not equal. For \( x = -2 \): \( f(-2) = -0.5(-2) + 2 = 1 + 2 = 3 \), \( g(-2) = 2(-2) - 3 = -4 - 3 = -7 \). Not equal. For \( x = 2 \): \( f(2) = 1 \), \( g(2) = 1 \). So the answer is 2. Wait, but the options include 2. So the correct input is 2.

Answer:

2