Problem 19 - 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: Testing for an even function

Substitute \(-x\) for \(x\) in \(g(x)\) and simplify. Therefore, \(g(-x)=(-x)^{2}+(-x)=x^{2}-x\).

Step 2: Comparing with original function for even property

Comparing \(g(-x)\) to \(g(x)\), one would see they are not equal. Hence, \(g(x)\) is not an even function.

Step 3: Testing for an Odd function

Take \(g(x)\) and replace \(x\) with \(-x\) and see if it's the negation of original function. We got \(g(-x)\) as \(x^{2}-x\) in step 1. This is not the negation of \(g(x)\).

Step 4: Concluding the function type

Since \(g(x)\) neither satisfies the conditions for an even nor odd function hence it's neither even nor odd.

Key Concepts

Function Tests

To determine if a function is even, odd, or neither, we perform a series of function tests. These tests involve substituting

• Replacing every instance of the variable in the function with \(-x\), checking for evenness,
• And examining if the output equals the original or its negation.

Firstly, the even function test: We say a function is even if replacing \(x\) with \(-x\) keeps the output exactly the same as the original. Algebraically, for a function \(f(x)\), it's even when \(f(-x) = f(x)\).

The odd function test, on the other hand, requires that \(f(-x)\) becomes the negative of \(f(x)\), i.e. \(f(-x) = -f(x)\).

If neither of these conditions hold true, the function is categorized as neither even nor odd.

Substitution Method

The substitution method in testing functions revolves around systematically replacing \(x\) with \(-x\) in the function.

In our example of \(g(x) = x^2 + x\), we substitute \(-x\) for \(x\), leading us to calculate \(g(-x) = (-x)^2 + (-x) = x^2 - x\).

We'll then examine \(g(-x)\) to see if it matches \(g(x)\) for evenness, or \( -g(x) \) for oddness.

• If \(g(-x) = g(x)\), then the function is even.
• If \(g(-x) = -g(x)\), then the function is odd.

Substituting in this manner simplifies the testing for symmetry about the y-axis and origin.

Algebraic Functions

Algebraic functions like \(g(x) = x^2 + x\) are functions built from variables and constants using algebraic operations.

Polynomial functions are common types of algebraic functions. They can manifest various symmetries depending on their terms.

• If all terms have even powers, they tend to be even functions.
• If all terms have odd powers, they might be odd functions.

In \(g(x) = x^2 + x\), there's a mix of an even powered term \(x^2\) and an odd powered term \(x\), leading to neither symmetry aligning the function as even nor odd.

Function Symmetry

Function symmetry is an essential concept in identifying even and odd functions. Symmetry can simplify graphing and understanding function behavior.

Even functions exhibit symmetry about the y-axis. This means if you fold the graph along the y-axis, both halves replicate each other perfectly.

Odd functions, in contrast, display origin symmetry, which involves a 180-degree rotational symmetry around the origin. This means turning the graph upside down reflects the same graph.

For \(g(x) = x^2 + x\), examining \(g(-x)\) results in \(x^2 - x\), showing neither y-axis nor origin symmetry, confirming it as neither even nor odd.