To determine the type of function represented by the table of values, we will analyze the provided data:
- The table gives \( x \) values: 1, 2, 3, 4, 5
- Corresponding \( f(x) \) values: 5376, 6720, 8400, 10500, 13125
Step 1: Check for Patterns
Let's look at the differences between the consecutive \( f(x) \) values.
- \( f(2) - f(1) = 6720 - 5376 = 1344 \)
- \( f(3) - f(2) = 8400 - 6720 = 1680 \)
- \( f(4) - f(3) = 10500 - 8400 = 2100 \)
- \( f(5) - f(4) = 13125 - 10500 = 2625 \)
Now, let's observe the differences of the differences to see if it can help us identify the function type (i.e., linear, quadratic, polynomial, exponential, etc.):
- Second differences:
- \( 1680 - 1344 = 336 \)
- \( 2100 - 1680 = 420 \)
- \( 2625 - 2100 = 525 \)
The second differences are not constant, which suggests that the function is not linear.
Step 2: Analyze Ratio of Consecutive Terms
To identify a possible multiplicative pattern or exponential growth, let's calculate the ratios of \( f(x) \) values:
- \( \frac{f(2)}{f(1)} = \frac{6720}{5376} \approx 1.25 \)
- \( \frac{f(3)}{f(2)} = \frac{8400}{6720} \approx 1.25 \)
- \( \frac{f(4)}{f(3)} = \frac{10500}{8400} \approx 1.25 \)
- \( \frac{f(5)}{f(4)} = \frac{13125}{10500} \approx 1.25 \)
Conclusion
The ratios of consecutive \( f(x) \) values are approximately constant (around 1.25), indicating that this function has a characteristic of exponential growth. Since it appears that every increase in \( x \) results in multiplication by a consistent factor of approximately 1.25, we can conclude that:
This function is exponential because the ratios of consecutive outputs are constant, implying a multiplicative growth pattern.