To determine if a function is linear or exponential, we typically look at how the function values change as the input (x) changes.
- A linear function has a constant rate of change: the differences between successive output values (f(x)) remain constant as x increases.
- An exponential function has a constant multiplicative rate of change: the ratio of successive output values remains constant as x increases.
Given the provided options, the first step would be to analyze the table of values, which isn't provided here. However, I can guide you on how to check whether a function is linear or exponential using typical criteria.
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For Linear:
- Calculate the differences between consecutive values f(x).
- If the differences are constant (say, 2 or 3), then the function is linear.
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For Exponential:
- Calculate the ratios of consecutive values f(x).
- If the ratios are constant (for example, a factor of 3), then the function is exponential.
Based on the responses you provided:
- "The function is linear because as x increases, f(x) increases additively by a difference of 2."
- "The function is linear because as x increases, f(x) increases additively by a difference of 3."
- "The function is exponential because as x increases, f(x) increases multiplicatively by a factor of 3."
- "The function is exponential because as x increases, f(x) increases multiplicatively by a factor of -1."
You would need to confirm whether the differences (for linear) or ratios (for exponential) are constant and match the statements given. If you determine that the successive differences are either 2 or 3, then the first two options are correct and the function is linear. If the ratios confirm a multiplication by 3 or -1, then the function is exponential.
Please analyze the table of values following the guidelines above to conclude correctly.