IB Mathematics AA HL Odd & even functions Study Notes
IB Mathematics AA HL Odd & even functions Study Notes
IB Mathematics AA HL Odd & even functions Study Notes Offer a clear explanation of Odd & even functions , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Odd & even functions.
Odd, Even, and Periodic Functions
Odd, Even, and Periodic Functions
Even function: A function is even if \( f(-x) = f(x) \) for all \( x \) in the domain. Graph is symmetric about the y-axis.
Odd function: A function is odd if \( f(-x) = -f(x) \) for all \( x \) in the domain. Graph is symmetric about the origin.
Neither: Some functions are neither odd nor even (e.g. \( f(x) = x^2 + x \)).
Periodic function: A function is periodic if there exists a constant \( T \) such that \( f(x + T) = f(x) \) for all \( x \). \( T \) is called the period.
Example :
Consider \( f(x) = \cos(x) \).Test the function type.
▶️Answer/Explanation
Consider \( f(x) = \cos(x) \)
Test: \( f(-x) = \cos(-x) = \cos(x) = f(x) \)
Conclusion: \( f(x) = \cos(x) \) is an even function. Its graph is symmetric about the y-axis.
Example :
Consider \( f(x) = \sin(x) \) . Test the function type.
▶️Answer/Explanation
Consider \( f(x) = \sin(x) \)
Test: \( f(-x) = \sin(-x) = -\sin(x) = -f(x) \)
Conclusion: \( f(x) = \sin(x) \) is an odd function. Its graph has rotational symmetry about the origin.
Example :
Consider \( f(x) = x^2 + x \).Test the function type.
▶️Answer/Explanation
Consider \( f(x) = x^2 + x \)
Test: \( f(-x) = (-x)^2 + (-x) = x^2 – x \)
\( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \)
Conclusion: \( f(x) = x^2 + x \) is neither even nor odd.
Finding the Inverse Function
Finding the Inverse Function
The inverse function \( f^{-1}(x) \) undoes the action of \( f(x) \). That is:
\( f^{-1}(f(x)) = x \quad \text{and} \quad f(f^{-1}(x)) = x \)
Key Steps to Find \( f^{-1}(x) \):
- Write \( y = f(x) \).
- Interchange \( x \) and \( y \).
- Solve for \( y \) in terms of \( x \).
- Write \( f^{-1}(x) \) for the expression of \( y \).
Domain Restriction: If \( f(x) \) is not one-to-one (fails the horizontal line test), we must restrict the domain so that \( f(x) \) has an inverse.
Reflection Property: The graph of \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) in the line \( y = x \).
Important: The domain of \( f^{-1}(x) \) is the range of \( f(x) \), and vice versa.
Example:
Find the inverse of \( f(x) = x^2 + 2 \) for \( x \ge 0 \).
▶️Answer/Explanation
Function \( y = x^2 + 2 \).
Swap \( x \) and \( y \): \( x = y^2 + 2 \).
Solve for \( y \):
\( x – 2 = y^2 \)
\( y = \sqrt{x – 2} \) (since \( x \ge 0 \), we choose the positive root)
Therefore, \( f^{-1}(x) = \sqrt{x – 2} \).
Domain of \( f^{-1}(x) \): \( x \ge 2 \) (since the range of \( f(x) \) is \( y \ge 2 \))
Self-Inverse Functions
Self-Inverse Functions
A self-inverse function is a function that is its own inverse. That means:
\( f(f(x)) = x \quad \forall x \in \text{domain of } f \)
Key properties of self-inverse functions:
- The graph of a self-inverse function is symmetrical about the line \( y = x \).
- Applying the function twice returns the original input value.
- The function and its inverse have identical rules (or expressions).
Visual interpretation: The graph of a self-inverse function overlaps its inverse when reflected across \( y = x \).
Example
Show that \( f(x) = \frac{1}{x} \), \( x \ne 0 \), is a self-inverse function.
▶️Answer/Explanation
check \( f(f(x)) = x \).
apply \( f \) to \( x \):
\( f(x) = \frac{1}{x} \)
Now apply \( f \) again to \( f(x) \):
\( f(f(x)) = f\left( \frac{1}{x} \right) = \frac{1}{\frac{1}{x}} = x \)
Therefore, \( f(x) = \frac{1}{x} \) is self-inverse.
Graphical note: The graph of \( f(x) = \frac{1}{x} \) is symmetric about the line \( y = x \).