Problem 47 - 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 \(f\). $$ f(x)=2 \sqrt{x} $$

Step-by-Step Solution

Answer

The function \(f(x) = 2\sqrt{x}\) is nonlinear.

Step 1: Identify the Function Type

To determine if the function is linear, consider the general form of a linear function, which is \(f(x) = ax + b\). A function is linear if it can be written in this form. Examine the given function, \(f(x) = 2\sqrt{x}\), and note that it involves a square root of \(x\), which means it cannot be written in the form \(ax + b\). Therefore, \(f(x)\) is not linear.

Step 2: Sketch the Graph of the Function

Let's graph \(f(x) = 2\sqrt{x}\) to visualize its behavior. Plot a few key points, such as:- When \(x = 0\), \(f(x) = 0\).- When \(x = 1\), \(f(x) = 2\).- When \(x = 4\), \(f(x) = 4\).The graph is a curve starting from (0,0) and increases as \(x\) increases, forming part of a sideways parabola. This demonstrates visually that \(f(x)\) is nonlinear, as a linear function would appear as a straight line on the graph.

Step 3: Determine if the Function is Constant

A constant function is a specific type of linear function where the value of \(f(x)\) is the same for all \(x\). Since \(f(x) = 2\sqrt{x}\) is not linear to begin with, it cannot be a constant function either. Furthermore, the values of \(f(x)\) change with \(x\), indicating that it is not constant.

Key Concepts

Linear Function

A linear function is a fundamental concept in mathematics, which might sound complex, but it's quite simple. Think of it like a straight line on a piece of paper. Its general form is given by the equation \( f(x) = ax + b \), where \( a \) and \( b \) are constants and \( x \) represents the variable or input. This equation tells us how \( f(x) \) changes as \( x \) changes. If you were to graph it, you'd see a straight line, making it predictable and easy to understand.

**Characteristics of Linear Functions**:

• The graph is always a straight line.
• The rate of change is constant. That means if you increase \( x \) by the same amount each time, \( f(x) \) will increase by a consistent amount.
• You can easily calculate the slope (which is \( a \) in \( f(x) = ax + b \)) and y-intercept (which is \( b \)).

Linear functions are everywhere in real life, from calculating the cost of goods to speed-time relationships. They're predictable because their graphs are straight lines, giving them a constant rate of change.

Constant Function

A constant function is a special kind of linear function. Imagine a horizontal line on a graph. No matter how far you go along the \( x \)-axis, the value of \( f(x) \) remains the same. This is a constant function. Its mathematical form is \( f(x) = c \), where \( c \) is a constant value.

**Characteristics of Constant Functions**:

• The graph is a horizontal line because the function value doesn’t change with \( x \).
• Every input \( x \) results in the same output \( f(x) \).
• The slope of the graph is zero, which means there’s no vertical change as \( x \) changes.

Constant functions are simple yet powerful in many real-world applications. For example, in fixed interest loans, the amount of interest paid each period can be modeled as a constant function.

Graphing Functions

Graphing functions is a visual way to understand the behavior of mathematical functions. By plotting points on a graph, you can see how a function behaves and changes as the variable \( x \) varies. This helps in understanding whether a function is linear, nonlinear, or constant.

**Tips for Graphing Functions**:

• Start by creating a table of values: Choose several \( x \)-values and calculate the corresponding \( f(x) \)-values.
• Plot these \( (x, f(x)) \) points on a graph.
• Join the points with a smooth curve or straight line depending on the function type.

A nonlinear function like \( f(x) = 2\sqrt{x} \) would not produce a straight line but a curve. This visual representation can easily show that \( f(x) \) is nonlinear because of the curve it forms. Graphing provides insight that pure numbers might not readily display and is a valuable tool in both learning and applying mathematics.