To determine if the function [tex]\( h(x) = 3x^3 + 4x \)[/tex] is even, odd, or neither, we will use the definitions of even and odd functions:
1. Even Function: A function [tex]\( f(x) \)[/tex] is even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
2. Odd Function: A function [tex]\( f(x) \)[/tex] is odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's proceed step-by-step to check the nature of the function.
### Step 1: Calculate [tex]\( h(-x) \)[/tex]
Substitute [tex]\( -x \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( h(x) = 3x^3 + 4x \)[/tex]:
[tex]\[ h(-x) = 3(-x)^3 + 4(-x) \][/tex]
### Step 2: Simplify [tex]\( h(-x) \)[/tex]
Calculate each term:
[tex]\[ 3(-x)^3 = 3(-x^3) = -3x^3 \][/tex]
[tex]\[ 4(-x) = -4x \][/tex]
So,
[tex]\[ h(-x) = -3x^3 - 4x \][/tex]
### Step 3: Compare [tex]\( h(-x) \)[/tex] with [tex]\( h(x) \)[/tex]
- Recall that [tex]\( h(x) = 3x^3 + 4x \)[/tex].
- [tex]\( h(-x) = -3x^3 - 4x \)[/tex].
### Step 4: Check if [tex]\( h(x) \)[/tex] is even
For [tex]\( h(x) \)[/tex] to be even, [tex]\( h(-x) \)[/tex] should be equal to [tex]\( h(x) \)[/tex]:
[tex]\[ 3x^3 + 4x \stackrel{?}{=} -3x^3 - 4x \][/tex]
Clearly, this is not true, so the function is not even.
### Step 5: Check if [tex]\( h(x) \)[/tex] is odd
For [tex]\( h(x) \)[/tex] to be odd, [tex]\( h(-x) \)[/tex] should be equal to [tex]\(-h(x)\)[/tex]:
[tex]\[ -h(x) = -(3x^3 + 4x) = -3x^3 - 4x \][/tex]
[tex]\[ h(-x) = -3x^3 - 4x \][/tex]
Since [tex]\( h(-x) = -h(x) \)[/tex] holds true, the function [tex]\( h(x) \)[/tex] is odd.
### Conclusion
The function [tex]\( h(x) = 3x^3 + 4x \)[/tex] is odd.
The correct answer is: Odd.
