UpStudy Free Solution:

To determine whether a graphed function is linear, we need to check if the graph forms a straight line. A linear function has a constant rate of change, meaning that the difference between successive outputs is the same for each unit increase in the input.

Let's evaluate each statement:

1. Yes, because each input value corresponds to exactly one output value.

- This is a characteristic of a function, but not specifically a linear function. Non-linear functions can also have each input corresponding to exactly one output value.

2. Yes, because the outputs increase as the inputs increase.

- This describes a positive correlation, but it does not confirm that the function is linear. Non-linear functions can also have outputs that increase as inputs increase.

3. No, because the graph is not continuous.

- Continuity is not a requirement for linearity. There can be linear functions that are not continuous (e.g., piecewise linear functions).

4. No, because the curve indicates that the rate of change is not constant.

- This is the correct reason. If the graph is a curve, it indicates that the rate of change is not constant, which means the function is not linear.

Therefore, the correct answer is:

No, because the curve indicates that the rate of change is not constant.

Supplemental Knowledge:

A linear function is a function whose graph is a straight line. This means that the rate of change between any two points on the graph is constant. The general form of a linear function is \( f(x) = mx + b \), where \( m \) represents the slope (rate of change) and \( b \) represents the y-intercept.

Key characteristics of linear functions include:

1. Constant Rate of Change: The difference in the output values divided by the difference in input values (slope) remains constant.

2. Straight Line Graph: The graph will always be a straight line.

3. One-to-One Correspondence: Each input value corresponds to exactly one output value, but this alone does not confirm linearity.

Nonlinear functions, on the other hand, have graphs that are not straight lines. These graphs can be curves, parabolas, hyperbolas, etc., indicating that the rate of change is not constant.

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