Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Sketch the graph of the function and determine whether the function is even, odd, or neither. \(f(x)=3\)

Step-by-Step Solution

Answer

The function \(f(x)=3\) is an even function, as the graph is a horizontal line at \(y=3\) which is symmetric about the y-axis.

Step 1: Sketching the graph

When the function is \(f(x)=3\), regardless of the value of x, the output is always 3. So, it will be a straight horizontal line passing through the point \(y=3\). Plot this on a 2-dimensional graph.

Step 2: Identifying Symmetries

An even function is one for which \(f(-x) = f(x)\) for all x, that is, reflecting the plot about the y-axis results in the same shape (symmetry about the y-axis). An odd function is one for which \(f(-x) = -f(x)\) for all x, meaning rotation of the plot by 180 degrees gives the same result (symmetry about the origin). In this case, for the function \(f(x)=3\), the plot is symmetric about the y-axis, confirming that it's an even function.

Key Concepts

Even and Odd Functions

Understanding the difference between even and odd functions is crucial in graphing. An *even function* follows the rule:

• \(f(x) = f(-x)\) for every value of \(x\).

This means that its graph is symmetrical about the y-axis. **Think mirror**: if you imagine a mirror on the y-axis, both sides of the graph look identical.

An *odd function* follows a different rule:

• \(f(-x) = -f(x)\) for every value of \(x\).

For these graphs, if you rotate the graph 180 degrees around the origin, it looks the same as the original. A simple example of an even function is \(f(x) = x^2\) and an odd function is \(f(x) = x^3\).

For the specific function \(f(x) = 3\), note that no matter whether you replace \(x\) with \(-x\), the value of the function remains 3. Thus, \(f(x) = f(-x)\), indicating that it's an even function.

Horizontal Line Graph

A horizontal line graph is one of the simplest types of graphs you can encounter. This happens when a function results in the same output for all input values. In the case of a constant function like \(f(x) = 3\), the graph is a straight, horizontal line that crosses the y-axis at the value 3.

This means:

• No matter what input \(x\) you choose, the output \(f(x)\) is always 3.
• The line is parallel to the x-axis.

Horizontal lines are particularly simple to sketch. You just need to find the y-value, which remains unchanged, and draw the line straight across the graph. Constant functions highlight the conceptual simplicity of having the same outcome irrespective of the input changes.

Function Symmetry

Function symmetry adds a delightful simplicity to understanding graphs. Symmetry can help detect patterns and make sketching graphs much easier. There are two main types of function symmetry you often encounter: symmetry about the y-axis and symmetry about the origin.

For symmetry about the y-axis:

• Imagine drawing a line straight down the y-axis. If the graph looks the same on both sides, then the function is even.

Functions exhibiting this type of symmetry include even functions which inherently follow the rule \(f(x) = f(-x)\).

Meanwhile, for symmetry about the origin:

• If flipping the graph upside-down or rotating it 180 degrees around the origin gives you the same graph, it has origin symmetry, and these are odd functions where \(f(-x) = -f(x)\).

Understanding these principles can vastly simplify your analysis of graphs, especially since any indication of symmetry narrows down the characteristics of the function you're dealing with. For the function \(f(x) = 3\), recognizing its symmetry about the y-axis helps you easily identify it as an even function.