Problem 15 - Precalculus : Building Concepts and Connections

Question

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=x^{4}+2 x^{2}-1$$

Step-by-Step Solution

Answer

The function \(g(x)=x^{4}+2 x^{2}-1\) is an even function and it portrays symmetry about the y-axis.

Step 1: Recognize the Problem Type

The task is about classification of a function based on its symmetry. The given function \(g(x)=x^{4}+2 x^{2}-1\) is a polynomial function.

Step 2: Validate Function as Even

Start with transforming \(x\) into \(-x\) in the function \(g(x)\). This gives \(g(-x)=(-x)^{4}+2(-x)^{2}-1=x^{4}+2x^{2}-1=g(x)\). Because \(g(-x) = g(x)\), the function is classified as even.

Step 3: Check Function for Oddness

Also, change \(x\) into \(-x\) again, but this time pay attention if \(g(-x)\) becomes \(-g(x)\). However, since \(g(-x)\) was found to be equal to \(g(x)\), it's clear that \(g(-x)\) will not be \(-g(x)\), thus the function is not odd.

Step 4: Understanding the Symmetry

Since we have established that the function is even, it can be concluded that it's symmetric with respect to the y-axis.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to powers, combined with coefficients. The general form of a polynomial function is expressed as \[a_n x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\]where each \(a_i\) is a coefficient and \(n\) is a non-negative integer that represents the degree of the polynomial.
Polynomial functions can vary in complexity, but they all share one key trait: the operations involving the variable \(x\) are restricted to addition, subtraction, multiplication, and non-negative integer exponentiation.

• A polynomial function of degree 0 is a constant function.
• Degree 1 corresponds to linear functions.
• Degree 2 is known as a quadratic function.
• The higher the degree, the more complex the curve.

Understanding the structure of polynomial functions helps in analyzing their symmetry and classification as odd or even.

Even Function

An even function is a type of function where swapping the input value with its negative does not change the output of the function. Mathematically, this is expressed as
\[f(-x) = f(x)\]
for all values of \(x\) in the domain of the function.
A simple way to remember this is: even functions "mirror" themselves around the y-axis.
For example, the function \(g(x) = x^4 + 2x^2 - 1\) is even because substituting \(-x\) for \(x\) gives us
\[g(-x) = (-x)^4 + 2(-x)^2 - 1 = x^4 + 2x^2 - 1 = g(x)\].
Evenness is a neat feature because it simplifies solving problems related to these functions, as the behavior is predictable and consistent on both sides of the y-axis.

Symmetry with Respect to Y-axis

Symmetry with respect to the y-axis occurs when the graph of a function looks the same on both sides of the y-axis. This means that for a given point \((x, y)\) on the graph, the point \((-x, y)\) will also be on it. This type of symmetry is a hallmark of even functions.
The graph of an even function like \(g(x) = x^4 + 2x^2 - 1\) clearly demonstrates this symmetry.

• If a function is symmetric with respect to the y-axis, it indicates that it is even.
• The behavior and shape of the graph repeat themselves on both left and right sides.

This symmetry makes it easier to analyze and predict the behavior of the function without needing to evaluate it across the entire domain, providing convenience in both graphical and analytical contexts.