Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Dec 06, 2023

Determine the parity of the function $g(x) = -2x^4 + 5x^2$ .

STEP 1

Assumptions
1. The function given is $g(x) = -2x^4 + 5x^2$ .
2. We need to determine whether the function is even, odd, or neither.
3. A function $f(x)$ is even if $f(x) = f(-x)$ for all $x$ in the domain of $f$ .
4. A function $f(x)$ is odd if $-f(x) = f(-x)$ for all $x$ in the domain of $f$ .
5. If a function is neither even nor odd, then it does not satisfy either of the two conditions above.

STEP 2

To determine if the function is even, we will substitute $-x$ for $x$ in the function and see if we get the original function back.
$g(-x) = -2(-x)^4 + 5(-x)^2$

STEP 3

Simplify the expression by calculating the powers of $-x$ .
$g(-x) = -2x^4 + 5x^2$

STEP 4

Compare the simplified expression of $g(-x)$ with the original function $g(x)$ .
$g(-x) = -2x^4 + 5x^2 = g(x)$

STEP 5

Since $g(-x) = g(x)$ , the function $g(x)$ is even.
The function $g(x) = -2x^4 + 5x^2$ is an even function.

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