To evaluate the function \( f(x) = 3 - 2x \), we can analyze it to determine which input results in the largest output.
This function is a linear equation in the form \( y = mx + b \), where \( m = -2 \) (the slope) and \( b = 3 \) (the y-intercept). Since the slope is negative, the function will decrease as \( x \) increases. This means that the function will generate larger outputs for smaller values of \( x \).
To find the input that generates the largest output, we can consider behavior at different points:
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As \( x \) approaches negative infinity (very large negative values), the function \( f(x) \) will yield higher outputs (since subtracting a negative value will increase the total).
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Specifically, we can evaluate the function at a few potential points:
- For \( x = 0 \): \[ f(0) = 3 - 2(0) = 3 \]
- For \( x = 1 \): \[ f(1) = 3 - 2(1) = 3 - 2 = 1 \]
- For \( x = -1 \): \[ f(-1) = 3 - 2(-1) = 3 + 2 = 5 \]
From this evaluation, we can see that:
- \( f(-1) = 5 \)
- \( f(0) = 3 \)
- \( f(1) = 1 \)
Since \( x = -1 \) gives the highest output among the tested values, and the output continues to increase without bound as \( x \) decreases, we conclude that the largest output occurs as \( x \) approaches negative infinity.
In conclusion, the largest output occurs with a very small (negative) input value, theoretically reaching infinity as \( x \) approaches negative infinity. If considering practical values, \( x = -1 \) generates a large output of 5, making it a reasonable choice.
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