Problem 45 - 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 \(f\). $$ f(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

The function \(f(x)=x^2-1\) is nonlinear and not a constant function.

Step 1: Assess the Power of the Variable

The function given is \(f(x) = x^2 - 1\). Notice that the term \(x^2\) is the highest power of the variable \(x\). This indicates that the function is quadratic because the highest exponent is 2, not 1.

Step 2: Define Linear Function Characteristics

A linear function is of the form \(f(x) = ax + b\), where the highest power of \(x\) is 1. Linear functions graph as straight lines, which have a constant slope.

Step 3: Compare the Function to Linear Form

In \(f(x) = x^2 - 1\), the term with the highest power of \(x\) is not 1 but 2. So, it does not match the form \(ax + b\). This tells us \(f\) is not a linear function.

Step 4: Graph the Function

To verify, we graph \(f(x) = x^2 - 1\), which results in a parabola opening upward with the vertex at (0, -1). A parabola graph confirms it is not a straight line.

Step 5: Draw Conclusions on Linearity and Constancy

Since \(f(x) = x^2 - 1\) is a quadratic and not a straight line, it is nonlinear. Consequently, since it's not a linear function, it cannot be a constant function either, as constant functions are a subset of linear functions with no \(x\) terms.

Key Concepts

Linear Functions

Linear functions are one of the simplest types of functions in algebra. A linear function is a function of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants, and \( x \) is the variable. This form ensures that when graphed, the function produces a straight line. The characteristic feature of linear functions is their slope, determined by the coefficient \( a \). This slope remains constant across the graph, hence the name "linear."
Linear functions are used widely in the modeling of real-world situations where relationships are directly proportional or consistently steady. For example, if you're traveling at constant speed, the distance over time can be described by a linear function.
In contrast, for nonlinear functions, such as quadratic functions, the graph is not a straight line but could be a curve. In the case of our exercise with \( f(x) = x^2 - 1 \), the presence of the squared term \( x^2 \) indicates it is nonlinear.

Constant Functions

Constant functions are a special type of linear function. They are characterized by having no variable part; the function is always equal to a single value. In mathematical terms, a constant function can be written as \( f(x) = c \), where \( c \) is a constant.
These functions are represented graphically by horizontal lines on the coordinate plane. Since the value of the function does not change with different inputs of \( x \), the line has a slope of zero. This is why constant functions are considered a subset of linear functions.
For our exercise, if a function like \( f(x) = x^2 - 1 \) were constant, it would not contain any \( x \) terms and would graph as a flat line. However, since our function includes \( x^2 \), it is neither linear nor constant.

Graphing Functions

Graphing functions is a key aspect of understanding their behavior visually. Using graphs, we can observe the relationship between the input (\( x \)) and the output (\( f(x) \)).
For linear functions, this means graphing a straight line, using key attributes like the slope and y-intercept. The slope gives us the angle and direction of the line, while the y-intercept is where the line crosses the y-axis.
Graphing quadratic functions, such as \( f(x) = x^2 - 1 \), involves plotting a curve called a parabola. In this example, the curve opens upwards because the \( x^2 \) term has a positive coefficient. The vertex of the parabola is a critical point, offering a minimum or maximum value of the function, which for \( f(x) = x^2 - 1 \) is at (0, -1).
Graphing allows us to confirm the type of function we're dealing with and visualize specific critical points such as intercepts and intervals of increase or decrease.