To determine the type of function based on the given table, we need to analyze the relationship between the values of \( x \) and their corresponding functional outputs \( f(x) \).
Given Data:
- \( x \): 11, 22, 33, 44, 55
- \( f(x) \): 22, 1313, 4646, 101101, 178178
Observations:
Let’s examine the output values \( f(x) \):
- For \( x = 11 \), \( f(11) = 22 \)
- For \( x = 22 \), \( f(22) = 1313 \)
- For \( x = 33 \), \( f(33) = 4646 \)
- For \( x = 44 \), \( f(44) = 101101 \)
- For \( x = 55 \), \( f(55) = 178178 \)
Now, we can look for a pattern:
- For \( x = 11 \): \( 2 \times 11 = 22 \)
- For \( x = 22 \): \( 60 \times 22 \approx 1313 \) - this significantly deviates from a simple multiplication.
- For \( x = 33 \): Similarly, \( 141 \times 33 \) or any fixed ratio doesn't yield a concise product.
Investigating \( f(x) \):
- The leading digits of each output seem to indicate a pattern, possibly related to concatenation rather than multiplication alone. The outputs appear to be increasing in magnitude rapidly.
- If we observe the functional growth—each output could also potentially be analyzed in the format \( f(x) \) as a polynomial or exponential function due to rapid increases in value.
Conclusion:
- The function \( f(x) \) appears to have a complex structure, likely a non-linear function due to the nature of the outputs. Each output is significantly larger than its predecessor, which aligns well with either a polynomial function with a high degree or an exponential function.
This function is likely non-linear because the differences between the outputs are not constant and grow larger, indicating a change in growth rate that aligns with polynomial or exponential behavior.
So, the final answer would be:
This function is non-linear because the values of \( f(x) \) increase at a rate that is not constant, indicating a more complex relationship than linear.