Question
Question Determine whether the function \( f(x)=-6 x^{7}-x^{5} \) is even, odd or neither. Answer odd even Examples neither
Ask by Boyd Bartlett. in the United States
Feb 26,2025
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Tutor-Verified Answer
Answer
The function \( f(x)=-6x^{7}-x^{5} \) is odd.
Solution
Analyze the function features with analysis_type="even_or_odd" and expression="f(x)=-6x^7-x^5"
Function by following steps:
- step0: Determine if even, odd or neither:
\(f\left(x\right)=-6x^{7}-x^{5}\)
- step1: Use substitution:
\(f\left(-x\right)=-6\left(-x\right)^{7}-\left(-x\right)^{5}\)
- step2: Simplify:
\(f\left(-x\right)=6x^{7}+x^{5}\)
- step3: Use substitution:
\(f\left(-x\right)=-f\left(x\right)\)
- step4: The function is an odd function:
\(\textrm{Odd}\)
The function \( f(x)=-6x^{7}-x^{5} \) is an odd function.
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Simplify this solution Extra Insights
To determine the nature of the function \( f(x) = -6x^7 - x^5 \), we can evaluate \( f(-x) \). If \( f(-x) = f(x) \), the function is even. If \( f(-x) = -f(x) \), then it is odd. Plugging in \( -x \) gives \( f(-x) = -6(-x)^7 - (-x)^5 = 6x^7 + x^5 \). Since \( f(-x) = -f(x) \), the function is odd. Another fun way to visualize odd functions is to think of them as having rotational symmetry about the origin. This means that if you take any point \( (x, f(x)) \) on the graph, the point \( (-x, -f(x)) \) will also be on the graph. So, for our function, if you plot points, you’ll see that they just flip as you cross through the origin—like a fun game of leapfrog!
