Odd & Even Functions
What are odd functions?
- A function is called odd if
- for all values of
- Examples of odd functions include:
- Power functions with odd powers: where
- For example:
- Some trig functions: , , ,
- For example:
- Linear combinations of odd functions
- For example:
What are even functions?
- A function is called even if
- for all values of
- Examples of even functions include:
- Power functions with even powers: where
- For example:
- Some trig functions: ,
- For example:
- Modulus function:
- Linear combinations of even functions
- For example:
What are the symmetries of graphs of odd & even functions?
- The graph of an odd function has rotational symmetry
- The graph is unchanged by a 180° rotation about the origin
- The graph of an even function has reflective symmetry
- The graph is unchanged by a reflection in the y-axis
Exam Tip
- Turn your GDC upside down for a quick visual check for an odd function!
- Ignoring axes, etc, if the graph looks exactly the same both ways, it's odd
Worked Example
a)
The graph is shown below. State, with a reason, whether the function is odd, even or neither.
b)
Use algebra to show that is an even function.
Periodic Functions
What are periodic functions?
- A function is called periodic, with period k, if
- for all values of
- Examples of periodic functions include:
- sin x & cos x: The period is 2π or 360°
- tan x: The period is π or 180°
- Linear combinations of periodic functions with the same period
- For example:
What are the symmetries of graphs of periodic functions?
- The graph of a periodic function has translational symmetry
- The graph is unchanged by translations that are integer multiples of
- The means that the graph appears to repeat the same section (cycle) infinitely
Exam Tip
- There may be several intersections between the graph of a periodic function and another function
- i.e. Equations may have several solutions so only answers within a certain range of -values may be required
- e.g. Solve for
- Alternatively you may have to write all solutions in a general form
- e.g.
- i.e. Equations may have several solutions so only answers within a certain range of -values may be required
Worked Example
The graph is shown below. Given that is periodic, write down the period.
Self-Inverse Functions
What are self-inverse functions?
- A function is called self-inverse if
- for all values of
- Examples of self-inverse functions include:
- Identity function:
- Reciprocal function:
- Linear functions with a gradient of -1:
What are the symmetries of graphs of self-inverse functions?
- The graph of a self-inverse function has reflective symmetry
- The graph is unchanged by a reflection in the line y = x
Exam Tip
- If your expression for is not the same as the expression for you can check their equivalence by plotting both on your GDC
- If equivalent the graphs will sit on top of one another and appear as one
- This will indicate if you have made an error in your algebra, before trying to simplify/rewrite to make the two expressions identical
- It is sometimes easier to consider self inverse functions geometrically rather than algebraically
Worked Example
Use algebra to show the function defined by is self-inverse.