Question
If \( f \) is an odd function, then \( f(-x)= \) The graph of an odd function is symmetric with respect to the If \( f \) is an odd function, then \( f(-x)= \) \( \begin{array}{l}x(x) . \\ -f(x)\end{array} \)
Ask by Alexander Ayala. in the United States
Feb 04,2025
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Tutor-Verified Answer
Answer
If \( f \) is an odd function, then \( f(-x) = -f(x) \). The graph of an odd function is symmetric with respect to the origin.
Solution
To solve the problem, we need to understand the properties of odd functions.
1. **Definition of an Odd Function**: A function \( f \) is called an odd function if it satisfies the condition:
\[
f(-x) = -f(x)
\]
for all \( x \) in the domain of \( f \).
2. **Symmetry of Odd Functions**: The graph of an odd function is symmetric with respect to the origin. This means that if you rotate the graph 180 degrees around the origin, it will look the same.
Now, let's summarize the answers to the questions:
- If \( f \) is an odd function, then:
\[
f(-x) = -f(x)
\]
- The graph of an odd function is symmetric with respect to the **origin**.
Thus, the final answers are:
- \( f(-x) = -f(x) \)
- The graph is symmetric with respect to the origin.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
If \( f \) is an odd function, then \( f(-x) = -f(x) \). This means that for every point on the graph at \( (x, f(x)) \), there is a corresponding point at \( (-x, -f(x)) \) which reflects across the origin. This property is what makes odd functions unique and intriguing! The graph of an odd function is symmetric with respect to the origin. If you were to rotate the graph 180 degrees around the origin, it would look unchanged! This characteristic helps in understanding the behavior of various functions, like sine and cubic functions, showcasing their beautiful symmetry in a visually appealing manner.
