Answer :
To determine whether the function [tex]\( w(x) = 3x^7 + 9x^3 \)[/tex] is even, odd, or neither, we need to follow a systematic approach.
1. Define an even function: A function [tex]\( f(x) \)[/tex] is considered even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
2. Define an odd function: A function [tex]\( f(x) \)[/tex] is considered odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's check if [tex]\( w(x) \)[/tex] is even or odd by substituting [tex]\( -x \)[/tex] into the function.
### Step-by-step Solution:
1. Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
w(-x) = 3(-x)^7 + 9(-x)^3
\][/tex]
2. Simplify the terms:
[tex]\[
(-x)^7 = -x^7 \quad \text{(since the exponent is odd)}
\][/tex]
[tex]\[
(-x)^3 = -x^3 \quad \text{(since the exponent is odd)}
\][/tex]
Substituting these back into the function:
[tex]\[
w(-x) = 3(-x^7) + 9(-x^3) = -3x^7 - 9x^3
\][/tex]
3. Compare [tex]\( w(-x) \)[/tex] with [tex]\( w(x) \)[/tex]:
[tex]\[
w(x) = 3x^7 + 9x^3
\][/tex]
[tex]\[
w(-x) = -3x^7 - 9x^3
\][/tex]
Notice that:
[tex]\[
w(-x) = -(3x^7 + 9x^3) = -w(x)
\][/tex]
4. Conclusion:
Since [tex]\( w(-x) = -w(x) \)[/tex], the function [tex]\( w(x) = 3x^7 + 9x^3 \)[/tex] is odd.
Therefore, the function [tex]\( w(x) = 3x^7 + 9x^3 \)[/tex] is odd.
1. Define an even function: A function [tex]\( f(x) \)[/tex] is considered even if [tex]\( f(-x) = f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
2. Define an odd function: A function [tex]\( f(x) \)[/tex] is considered odd if [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
Let's check if [tex]\( w(x) \)[/tex] is even or odd by substituting [tex]\( -x \)[/tex] into the function.
### Step-by-step Solution:
1. Substitute [tex]\(-x\)[/tex] into the function:
[tex]\[
w(-x) = 3(-x)^7 + 9(-x)^3
\][/tex]
2. Simplify the terms:
[tex]\[
(-x)^7 = -x^7 \quad \text{(since the exponent is odd)}
\][/tex]
[tex]\[
(-x)^3 = -x^3 \quad \text{(since the exponent is odd)}
\][/tex]
Substituting these back into the function:
[tex]\[
w(-x) = 3(-x^7) + 9(-x^3) = -3x^7 - 9x^3
\][/tex]
3. Compare [tex]\( w(-x) \)[/tex] with [tex]\( w(x) \)[/tex]:
[tex]\[
w(x) = 3x^7 + 9x^3
\][/tex]
[tex]\[
w(-x) = -3x^7 - 9x^3
\][/tex]
Notice that:
[tex]\[
w(-x) = -(3x^7 + 9x^3) = -w(x)
\][/tex]
4. Conclusion:
Since [tex]\( w(-x) = -w(x) \)[/tex], the function [tex]\( w(x) = 3x^7 + 9x^3 \)[/tex] is odd.
Therefore, the function [tex]\( w(x) = 3x^7 + 9x^3 \)[/tex] is odd.
