To determine if the function is linear, we need to check if the differences between the values of $y$ (dependent variable) are consistent as the values of $x$ (independent variable) change. In other words, for a function to be linear, the rate of change of $y$ with respect to $x$ should be constant.

Step 1: Check the differences in $y$

Given data points:

• $x = 10, 20, 30, 40, 50$
• $y = 32.9, 33.5, 34.1, 34.6, 35.2$

Calculate the differences in $y$ values between consecutive points:

$\Delta y_{1} = 33.5 - 32.9 = 0.6$ $\Delta y_{2} = 34.1 - 33.5 = 0.6$ $\Delta y_{3} = 34.6 - 34.1 = 0.5$ $\Delta y_{4} = 35.2 - 34.6 = 0.6$

Step 2: Determine if the differences are constant

The differences are not exactly constant. There is a difference of 0.5 between $y_3$ and $y_4$, while the others are all 0.6. This means the rate of change is not constant, so the function is not linear.

Conclusion:

The function is not linear. The correct answer is:

The function is not linear.

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Would you like to go through a related example? Here are some questions you might find helpful:

1. How do you calculate the slope of a linear function?
2. How can you tell if a function is linear from a graph?
3. What are the key differences between linear and nonlinear functions?
4. What are the steps for finding the equation of a linear function?
5. How do you calculate the slope and intercept of a line from a table of values?

Tip: When checking if a function is linear, always look for constant differences in the $y$-values when the $x$-values increase by the same amount.