To determine the nature of the function depicted in the table, let's first look at the pairs given:
- \( f(-3) = 1782 \)
- \( f(-2) = 970 \)
- \( f(-1) = 4950 \)
- \( f(0) = 8250 \)
- \( f(1) = 13750 \)
Now, we need to analyze how the outputs change as the inputs change.
- Identify the output values:
- At \( x = -3 \), the output is 1782.
- At \( x = -2 \), the output is 970. (Decreases)
- At \( x = -1 \), the output is 4950. (Increases)
- At \( x = 0 \), the output is 8250. (Increases)
- At \( x = 1 \), the output is 13750. (Increases)
Given the changes in the outputs, we can observe that the function does not follow a linear or simple polynomial form because the decrease at \( -2 \) and the variable increases suggest a more complex relationship.
Next, let's consider the possibility of a quadratic or exponential function. To go further, we can look at the ratios or check for a common pattern in differences, but without those calculations presented, we'll focus on defining the function type based on its behavior.
Given the values:
- The function values have non-consistent intervals and appear to be changing rapidly, hinting towards a polynomial or exponential nature.
To summarize:
This function is not linear because the changes in output do not correspond to a constant change as seen in a linear function. The outputs rapidly increase especially around zero.
Answer
This function is non-linear because the rate of change of the function values is not constant, and the input-output relationship does not fit a straight line (it involves varying increments).
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