Problem 60 - Contemporary Precalculus

Question

Give an example of a function \(g\) with the property that \(g(x)=g(-x)\) for every real number \(x\)

Step-by-Step Solution

Answer

Question: Provide an example of a function g that satisfies the property g(x) = g(-x) for every real number x. Answer: An example of a function g with the property g(x) = g(-x) for every real number x is g(x) = 3x^2.

Step 1: Understand the Given Information

Notice that the given property g(x) = g(-x) indicates that g is an even function. An even function is symmetric about the y-axis, meaning that its graph is unchanged if it is reflected across this axis.

Step 2: Example of an Even Function

A simple example of an even function is a quadratic function of the form g(x) = ax^2, where a is a non-zero real constant. When x is replaced by -x, g(-x) = a(-x)^2 = ax^2 = g(x). Therefore, any quadratic function of this form is an example of an even function satisfying the given property.

Step 3: Provide the Even Function Example

An example of a function g with the property g(x) = g(-x) for every real number x is g(x) = 3x^2, which is a quadratic function of the form g(x) = ax^2 and satisfies the given property.

Key Concepts

Function Symmetry

Function symmetry is a crucial concept in mathematics, especially when analyzing even and odd functions. When a function is symmetric, it has a balanced appearance across a particular line or point.

For even functions, like the one described in the original exercise, symmetry occurs over the y-axis. This means if you pick any point on the graph and reflect it across the y-axis, it will land on another point of the graph.

This symmetry property is mathematically expressed as:

• For every function value at a point \(x\), the function value at \(-x\) is the same, i.e., \(f(x) = f(-x)\).

Understanding this concept helps us visually identify and analyze the behavior of functions quickly.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2 and have the general form \(f(x) = ax^2 + bx + c\). They are among the simplest even functions when \(b = 0\), which ensures they are symmetric around the y-axis.

In our example, a specific form \(g(x) = ax^2\) shows an even quadratic function. Here:

• \(a\) is a constant that controls the width and direction of the parabola.
• The lack of a linear term \(bx\) assures perfect y-axis symmetry.

When \(g(x) = 3x^2\), we have a parabola opening upwards, and this function is an excellent demonstration of a quadratic that meets the symmetry condition \(g(x) = g(-x)\).

Graph Properties

Graph properties of quadratic functions reveal many insights into their structure and behavior. These functions typically graph as parabolas, which have distinct characteristics:

• Vertex: This is the highest or lowest point on the parabola, representing the function's maximum or minimum value.
• Axis of symmetry: For our specific quadratic of the form \(ax^2\), this line of symmetry is the y-axis itself.
• Direction of opening: Controlled by the coefficient \(a\), the parabola can open upwards (if \(a > 0\)) or downwards (if \(a < 0\)).

The function \(g(x) = 3x^2\) has these graph properties, making it very predictable and easy to work with in calculations and graphs.

Y-Axis Symmetry

Y-axis symmetry is a defining feature of even functions. When a function, like \(g(x) = 3x^2\), displays this symmetry, the graph appears identical on both sides of the y-axis.

This symmetry means:

• Reflecting the graph over the y-axis will not change its appearance.
• For every point \((x, g(x))\), there is a corresponding point \((-x, g(x))\).

This conceptual understanding of y-axis symmetry makes it easier to graph functions and notice patterns in their behavior. It allows for predicting the function's values just by knowing half of the graph, a useful property for both plotting by hand and analyzing data.

Function symmetry, especially y-axis symmetry, simplifies many mathematical problems and helps in the study of function behaviors and transformations.