Problem 20 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$g(x)=x^{2}-x$$

Step-by-Step Solution

Answer

The function \( g(x)=x^{2}-x \) is neither even nor odd.

Step 1: Evaluation of g(-x)

First, replace \( x \) with \( -x \) in \( g(x) = x^{2}-x \). The resulting function becomes \( g(-x)=(-x)^{2}-(-x) = x^{2}+x \)

Step 2: Comparing original function with g(-x)

Now, compare the results of \( g(-x) = x^{2}+x \) with the original function \( g(x)=x^{2}-x \). It can be seen that \( g(-x) \) does not match either \( g(x) \) or \( -g(x) \).

Step 3: Conclusion

Since \( g(-x) \) does not match either \( g(x) \) or \( -g(x) \), it is concluded that the function \( g(x)=x^{2}-x \) is neither even nor odd.

Key Concepts

Function Evaluation

To determine the characteristics of a function, the first step often involves function evaluation. This involves substituting different values for the variable, typically replacing it with its negative to check symmetry.
For the given function, we start with:

• Evaluating the original function: \( g(x) = x^{2} - x \).
• Replacing \( x \) with \( -x \): \( g(-x) = (-x)^2 - (-x) = x^2 + x \).

Calculating \( g(-x) \) provides a new function that can help us understand the symmetry properties of the original function.
This process of evaluating \( g(-x) \) is crucial in determining whether the function is even or odd.
Substitution like this is a basic yet powerful tool in dissecting the nature of functions.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of one variable with coefficients. They often appear in a form such as \( ax^n + bx^{n-1} + \, ... \, + c \).
In our case, the function \( g(x) = x^{2} - x \) is called a quadratic polynomial because it includes terms up to \( x^2 \).
Polynomial functions, like \( g(x) \), are generally easy to manipulate and evaluate due to their straightforward algebraic structure.

• Each term in a polynomial is made of coefficients and the variable to a non-negative power.
• The degree of the polynomial is dictated by the highest power of the variable.

Quadratic polynomials, in particular, have a distinct shape when graphed—a parabola—which can provide additional insights into their symmetry and behavior.
Understanding these basics can make working with polynomial functions more intuitive.

Symmetry in Functions

A function's symmetry determines whether it is even, odd, or neither, and discovering this involves evaluating specific properties. An even function satisfies \( f(-x) = f(x) \) for all \( x \) in the function's domain, resulting in a graph that is symmetric about the y-axis.
Conversely, odd functions satisfy \( f(-x) = -f(x) \), showing symmetry about the origin.

• An example of an even function is \( f(x) = x^2 \), where flipping the sign of \( x \) results in the same output.
• An example of an odd function is \( f(x) = x^3 \), where flipping the sign of \( x \) results in the output's sign being flipped as well.

For the function \( g(x)=x^{2}-x \), these symmetry conditions are not met:

• \( g(-x) = x^{2} + x \) is neither equal to \( g(x) = x^2 - x \) nor \( -g(x) = -x^2 + x \).

This evaluation indicates that \( g(x) \) is neither even nor odd.
Grasping the concepts of symmetry helps to predict and understand a function's behavior visually and algebraically.