To determine which graph accurately represents the function \( f(x) = \frac{1}{2}x + 3 \) using input-output tables, we first need to calculate the output values (f(x)) for the given inputs: −4, −2, 0, 2, and 4.
We will compute:
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For \( x = -4 \): \[ f(-4) = \frac{1}{2}(-4) + 3 = -2 + 3 = 1 \]
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For \( x = -2 \): \[ f(-2) = \frac{1}{2}(-2) + 3 = -1 + 3 = 2 \]
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For \( x = 0 \): \[ f(0) = \frac{1}{2}(0) + 3 = 0 + 3 = 3 \]
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For \( x = 2 \): \[ f(2) = \frac{1}{2}(2) + 3 = 1 + 3 = 4 \]
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For \( x = 4 \): \[ f(4) = \frac{1}{2}(4) + 3 = 2 + 3 = 5 \]
Now, we can summarize the input-output results in a table:
\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -4 & 1 \ -2 & 2 \ 0 & 3 \ 2 & 4 \ 4 & 5 \ \hline \end{array} \]
Now we have the following points:
- \( (-4, 1) \)
- \( (-2, 2) \)
- \( (0, 3) \)
- \( (2, 4) \)
- \( (4, 5) \)
To identify the correct graph, look for a graph that includes these points. The graph should be a straight line, as the function \( f(x) = \frac{1}{2}x + 3 \) is linear.
If you have multiple graph options, find the one that passes through all these specified points, and that will be the correct graph representing the function.
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