Question

Identify the odd function.(1 point)
Responses

f(x)=3x2
f left parenthesis x right parenthesis equals 3 x squared

f(x)=x4
f left parenthesis x right parenthesis equals x superscript 4 baseline

f(x)=2x3
f left parenthesis x right parenthesis equals 2 x cubed

f(x)=2(x+1)3

Answer 1

To determine which of the given functions is an odd function, we need to apply the definition of an odd function: A function \( f(x) \) is considered odd if \( f(-x) = -f(x) \) for all \( x \).

Let's evaluate each of the provided functions:

1. **\( f(x) = 3x^2 \)**:
\[
f(-x) = 3(-x)^2 = 3x^2 = f(x)
\]
This is an even function, not odd.

2. **\( f(x) = x^4 \)**:
\[
f(-x) = (-x)^4 = x^4 = f(x)
\]
This is also an even function, not odd.

3. **\( f(x) = 2x^3 \)**:
\[
f(-x) = 2(-x)^3 = -2x^3 = -f(x)
\]
This function is odd.

4. **\( f(x) = 2(x+1)^3 \)**:
\[
f(-x) = 2(-x+1)^3 = 2(1-x)^3 \quad \text{(needs further simplification)}
\]
This does not simplify directly to \(-f(x)\), so let’s calculate:
\[
f(-x) \neq -f(x) \quad \text{(After evaluating, it's determined that this function is neither odd nor even but not fully evaluated here.)}
\]

The only odd function from the ones given is **\( f(x) = 2x^3 \)**.