A coordinate grid with two lines. One line labeled f(x) and passes through (negative 3, 1), (0, 1), and point (1.5, 1). The second line is labeled g(x) and passes through point (0, negative 1), point (1.5, 1), and point (3, 3).

Question

What is the best approximation for the input value when f(x)=g(x)?

0.0
0.5
1.5
3.0

GPT-4o mini · Answer

To find the point where $ f(x) = g(x) $, we first identify the behavior of both functions based on the points provided. 1. **Function $ f(x) $**: - Points: $ (-3, 1) $, $ (0, 1) $, $ (1.5, 1) $ - From these points, we can see that $ f(x) = 1 $ for all $ x $ in the range given (it appears to be a horizontal line at $ y = 1 $). 2. **Function $ g(x) $**: - Points: $ (0, -1) $, $ (1.5, 1) $, $ (3, 3) $ - To determine the equation of $ g(x) $, we can find the slope between the points $ (0, -1) $ and $ (1.5, 1) $: $$ \text{slope} = \frac{1 - (-1)}{1.5 - 0} = \frac{2}{1.5} = \frac{4}{3} $$ - Using the point-slope form of the equation: $$ g(x) - (-1) = \frac{4}{3}(x - 0) $$ $$ g(x) = \frac{4}{3}x - 1 $$ Now, we set $ f(x) = g(x) $ to find the intersection: $$ 1 = \frac{4}{3}x - 1 $$ Adding 1 to both sides: $$ 2 = \frac{4}{3}x $$ Multiplying both sides by $ \frac{3}{4} $: $$ x = \frac{3}{2} = 1.5 $$ Thus, $ f(x) = g(x) $ at $ x = 1.5 $. Therefore, the best approximation for the input value when $ f(x) = g(x) $ is **1.5**.