Question

Determine whether each function is linear or nonlinear. If it is linear, determine the slope. $$ \begin{array}{|cc|} \hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\ \hline-2 & 1 / 4 \\ -1 & 1 / 2 \\ 0 & 1 \\ 1 & 2 \\ 2 & 4 \\ \hline \end{array} $$

Short answer

The function is nonlinear.

Step-by-step solution

Understand What Makes a Function Linear

A function is linear if it can be written in the form \( f(x) = mx + b \), where \(m\) and \(b\) are constants. This means that the function should have a constant rate of change, or slope.

Calculate Slopes Between Points

Calculate the slope between consecutive pairs of points using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Checking consecutive points:1. Between (-2, 1/4) and (-1, 1/2): \[m = \frac{1/2 - 1/4}{-1 - (-2)} = \frac{1/4}{1} = 1/4\]2. Between (-1, 1/2) and (0, 1): \[m = \frac{1 - 1/2}{0 - (-1)} = \frac{1/2}{1} = 1/2\]The slopes are not equal, indicating the function is nonlinear (the rate of change is not constant).

Confirm Nonlinearity

Since we found that the slopes between pairs of points are not constant, the function does not have a constant rate of change. Therefore, we can confirm that the function is nonlinear.

Key concepts

slope calculation

One of the key aspects to determine whether a function is linear or nonlinear is slope calculation. Slope essentially measures how steep a line is. If the slope is the same between any two points on the graph of a function, then the function is linearly consistent.
To calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \'m\' represents the slope.
Let's take two points from our example: (-2, 1/4) and (-1, 1/2). Plugging these values into the slope formula, we get:
\[ m = \frac{1/2 - 1/4}{-1 - (-2)} = \frac{1/4}{1} = 1/4 \]
If we compute the slope for another pair of points, say (-1, 1/2) and (0, 1), it gives us:
\[ m = \frac{1 - 1/2}{0 - (-1)} = \frac{1/2}{1} = 1/2 \]
Notice the slopes are different, suggesting the function is nonlinear.

constant rate of change

For a function to be linear, it must have a constant rate of change across all points. The rate of change in mathematics is often represented by its slope.
This means if a function is linear, the slope between any two points should always be the same. A constant slope signifies a straight line graph which fits the linear function form: \(\( f(x) = mx + b \)\).
In our given example, the slope changes from 1/4 to 1/2 and so on. This variation indicates that the rate of change is not consistent, causing the function to be classified as nonlinear.
Understanding the consistent rate of change is crucial for identifying linear properties, and any deviations confirm non-linearity.

nonlinear functions

Nonlinear functions are functions where the rate of change is not constant. This means the slope of the function changes as we move from point to point.
The given example is a clear depiction of a nonlinear function as the slopes between different points are different. Nonlinear functions usually form curves on a graph rather than a straight line.
Nonlinear behavior can be seen in various natural phenomena and hence, it is important not only for mathematics but also for applying these concepts in the real world.
So if you ever calculate slopes between points and they turn out different, you're dealing with a nonlinear function.