For each function below, determine whether it is Even, Odd, or Neither

f(x)=x2+4   Select an answer Odd Even Neither 

g(x)=x3+4   Select an answer Odd Even Neither 

h(x)=4x3+x   Select an answer Odd Even Neither 

To determine whether a function is even, odd, or neither, we use the following tests:

- **Even Function:** \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \).
- **Odd Function:** \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \).
- **Neither:** If neither of the above conditions hold, the function is neither even nor odd.

Now, let's analyze each function:

### 1. \( f(x) = x^2 + 4 \)
- To check if it's even:
  \[
  f(-x) = (-x)^2 + 4 = x^2 + 4
  \]
  Since \( f(-x) = f(x) \), the function is **even**.

### 2. \( g(x) = x^3 + 4 \)
- To check if it's odd:
  \[
  g(-x) = (-x)^3 + 4 = -x^3 + 4
  \]
  \( g(-x) \neq -g(x) \) and \( g(-x) \neq g(x) \), so the function is **neither**.

### 3. \( h(x) = 4x^3 + x \)
- To check if it's odd:
  \[
  h(-x) = 4(-x)^3 + (-x) = -4x^3 - x = -(4x^3 + x)
  \]
  Since \( h(-x) = -h(x) \), the function is **odd**.

### Final answers:
- \( f(x) = x^2 + 4 \) is **even**.
- \( g(x) = x^3 + 4 \) is **neither**.
- \( h(x) = 4x^3 + x \) is **odd**.

Would you like any further details or explanations?

### Relative Questions:
1. What is the geometric interpretation of an even function?
2. Can a function be both even and odd? Why or why not?
3. How do you determine if a function is neither even nor odd?
4. What kind of symmetry does an odd function exhibit on a graph?
5. How do polynomials' degrees affect whether they are even, odd, or neither?

**Tip:** If a function contains only even powers of \(x\), it tends to be even, and if it contains only odd powers, it tends to be odd.

For each function below, determine whether it is Even, Odd, or Neither: f(x)=x^2+4, g(x)=x^3+4, h(x)=4x^3+x

Learn how to determine whether functions are even, odd, or neither by analyzing f(x)=x^2+4, g(x)=x^3+4, and h(x)=4x^3+x. Understand key concepts like symmetry and function properties. This problem is suitable for students in Grades 10-12.

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