How do you determine if #g(x) = (4+x^2)/(1+x^4)# is an even or odd function?

Question

Anna Allen · Accepted Answer

even function

To determine if a function is even/odd the following applies. • If a function is even then f(x) = f(-x) , for all x Even functions are symmetrical about the y-axis • If a function is odd then f(-x) = - f(x) Odd functions have symmetry about the origin Test for even : # g(-x) = (4 + (-x)^2)/(1 + (-x)^4) = (4 + x^2)/(1 + x^4) = g(x)" hence even "# Here is the graph of the function. Note symmetry about y-axis. graph{(4+x^2)/(1+x^4) [-10, 10, -5, 5]}

Sadie Alexander · Answer

undefined To determine if a function $ g(x) = \frac{4 + x^2}{1 + x^4} $ is even or odd, we can use the properties of even and odd functions:

1. **Even Functions**: A function $ f(x) $ is even if $ f(-x) = f(x) $ for all values of $ x $.

2. **Odd Functions**: A function $ f(x) $ is odd if $ f(-x) = -f(x) $ for all values of $ x $.

First, let's evaluate $ g(-x) $ and $ -g(x) $ to see which property $ g(x) $ satisfies:

$$ g(-x) = \frac{4 + (-x)^2}{1 + (-x)^4} $$
$$ g(-x) = \frac{4 + x^2}{1 + x^4} $$

$$ -g(x) = -\frac{4 + x^2}{1 + x^4} $$

Comparing $ g(-x) $ and $ -g(x) $, we see that they are not equal, nor is one the negation of the other. Therefore, the function $ g(x) = \frac{4 + x^2}{1 + x^4} $ does not satisfy the criteria for an even or odd function. It is neither even nor odd.