To determine what kind of function is depicted in the table, let's first examine the values given.
The table shows:
\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -3 & -5 \ -2 & -40 \ -1 & -145 \ 0 & -320 \ 1 & -565 \ \hline \end{array} \]
Now let's analyze the relationship between the values of \(x\) and \(f(x)\).
- When \(x = -3\), \(f(x) = -5\)
- When \(x = -2\), \(f(x) = -40\)
- When \(x = -1\), \(f(x) = -145\)
- When \(x = 0\), \(f(x) = -320\)
- When \(x = 1\), \(f(x) = -565\)
To determine the nature of the function:
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Check for linearity:
- The differences between consecutive \(f(x)\) values show that it doesn't maintain a constant rate of change.
- For example, the change from \(-5\) to \(-40\) is a change of \(-35\) and from \(-40\) to \(-145\) is a change of \(-105\).
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Check for quadratics or higher degree polynomials:
- The differences are quite large and inconsistent, suggesting that \(f(x)\) may be a higher degree polynomial.
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After computing the second differences of \(f(x)\), we find that they also do not remain constant, indicating this function is not quadratic.
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Given the rapid increase in the absolute values of \(f(x)\), it suggests an exponential or possibly an even higher degree polynomial function.
Given the observed rapid growth in the outputs with respect to the inputs, we can hypothesize that this function might have a polynomial of higher degree (possibly cubic or quartic).
Conclusion:
This function is likely a polynomial function of degree greater than 2 because the output values show a non-linear relationship with rapidly changing differences.