Let's analyze the table of values from the uploaded image:

The table shows:

To determine if the function is linear, we need to check if the rate of change of $y$ with respect to $x$ (the slope) is constant.

Step 1: Calculate the rate of change (slope) between consecutive points

The slope between two points is given by:

$\text{slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Between (18, 9) and (19, 11):

$\text{slope} = \frac{11 - 9}{19 - 18} = \frac{2}{1} = 2$

Between (19, 11) and (20, 18):

$\text{slope} = \frac{18 - 11}{20 - 19} = \frac{7}{1} = 7$

Step 2: Compare the slopes

• The first slope is $2$, and the second slope is $7$.
• Since the slopes are not constant, the function is nonlinear.

Final Answer:

The function is nonlinear.

Would you like a detailed explanation of why constant slopes imply linearity?

Here are 5 related questions you might find helpful to explore:

1. What defines a linear function mathematically, and how can it be recognized from a table of values?
2. How can we graphically check if a function is linear or nonlinear?
3. What does a constant rate of change signify in real-world applications?
4. How do nonlinear functions differ in behavior compared to linear functions?
5. Can a function with varying slopes still represent a certain pattern (e.g., quadratic)?

Tip: To quickly determine linearity in a table, always check if the ratio of change in $y$ to $x$ remains constant for all intervals.