Problem 42 - College Algebra with Modeling and Visualization

Question

Determine if \(f\) is a linear or nonlinear function. If \(f\) is a linear function, determine if \(f\) is a constant function. Support your answer by graphing \(f\). $$ f(x)=-2 $$

Step-by-Step Solution

Answer

The function is linear and constant because it can be expressed as \(0x - 2\), leading to a horizontal line on the graph.

Step 1: Identify the Type of Function

The given function is \( f(x) = -2 \). Notice that this equation can be rewritten as \( f(x) = 0x - 2 \), which is in the form \( f(x) = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 0 \) and \( b = -2 \).

Step 2: Determine Linearity

Since the function \( f(x) = -2 \) can be expressed in the form \( mx + b \), it is a linear function. A linear function is represented as a straight line on a graph, characterized by a constant rate of change.

Step 3: Check if the Function is Constant

In the expression \( f(x) = 0x - 2 \), since the coefficient of \( x \), which is \( m \), is 0, indicating that the function does not change with \( x \). Therefore, the function is a constant function as it always returns the same output, \(-2\), for any input.

Step 4: Graph the Function

To graph the function, plot the constant value \( f(x) = -2 \) on the y-axis. The graph is a horizontal line that crosses the y-axis at \(-2\) and runs parallel to the x-axis, confirming it is constant.

Key Concepts

Constant Function

A constant function is a specific type of linear function. Its defining characteristic is that it always produces the same output, no matter what input you provide. In mathematical terms, a constant function is written as \( f(x) = c \) where \( c \) is a fixed number. For example, the function \( f(x) = -2 \) remains e -2 no matter what the value of \( x \) is.Key features of constant functions include:

• The slope \( m \) is zero. This means that the line does not tilt; it remains flat and horizontal.
• The graph is a straight line parallel to the x-axis.
• The y-intercept occurs at the constant value, which is where the line crosses the y-axis.

Understanding constant functions helps in recognizing their graphs and knowing that they indicate no change as \( x \) varies.

Graphing Functions

Graphing functions provides a visual representation of relationships between variables. For linear functions, like constant functions, this is particularly simple.To graph the function \( f(x) = -2 \), you would:

• Identify the y-value where the line sits, which is at \( y = -2 \).
• Draw a horizontal line across the graph at \( y = -2 \), which is a direct reflection of the function being constant.

Tips for graphing functions:

• Always start by identifying the y-intercept, which tells you where the function crosses the y-axis. For a constant function, this is its only feature.
• Since there's no rate of change, no slope needs to be considered for constant functions. The line will not ascend or descend, maintaining a steady position.

Graphing functions, especially constant ones, helps reinforce the understanding of how outputs remain uniform regardless of any changes to inputs.

Rate of Change

The rate of change is a fundamental concept in understanding how different types of functions behave. For linear functions, it's often represented as the slope of the line, denoted as \( m \) in the equation \( f(x) = mx + b \).Here's what you need to remember about the rate of change:

• In a constant function, the rate of change is zero because the function produces the same result regardless of the input. This is evident in the equation \( f(x) = 0x + b \).
• A zero rate of change results in a horizontal line on the graph, indicating that there's no increase or decrease as \( x \) changes.
• Recognizing the rate of change helps in predicting the behavior of a function, especially when comparing it with other types of linear functions that have non-zero slopes.

By understanding the rate of change, you can more easily identify the nature of a function and represent it graphically with accuracy.