determine whether this function is even odd or neither. y = x^4

To determine whether the function \( y = x^4 \) is even, odd, or neither, we'll follow these steps:

1. **Even Function Test:**
   A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain.

2. **Odd Function Test:**
   A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \) in its domain.

Now, let's apply these tests to the function \( y = x^4 \).

### Step 1: Test for Evenness
Calculate \( f(-x) \):
\[
f(-x) = (-x)^4 = (-x)(-x)(-x)(-x) = x^4
\]

Since \( f(-x) = f(x) \), the function is even.

### Step 2: Test for Oddness
Calculate \( -f(x) \):
\[
-f(x) = -(x^4) = -x^4
\]

Since \( f(-x) \) is not equal to \( -f(x) \), the function is not odd.

### Conclusion:
The function \( y = x^4 \) is **even**.

Would you like more details on this topic or have any questions?

Here are 8 related questions you might be interested in:
1. How do you determine if a function is odd if the function involves multiple terms?
2. What are some examples of functions that are neither even nor odd?
3. Can a function be both even and odd? If so, what kind of function would that be?
4. What is the significance of even and odd functions in calculus?
5. How do symmetry properties of functions relate to even and odd functions?
6. What happens to the evenness or oddness of a function if we add a constant to it?
7. How can we graphically identify if a function is even or odd?
8. What are the implications of even and odd functions in Fourier series?

**Tip:** For polynomial functions, if all the exponents of \( x \) are even, the function is even; if all are odd, the function is odd.

Explore how to determine whether the function y = x^4 is even, odd, or neither with detailed steps and explanations. Understand the tests for evenness and oddness of functions, applied specifically to the polynomial function y = x^4. Suitable for students in Grades 11-12.

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