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Integrated math 3
Course: Integrated math 3 > Unit 6
Lesson 3: Symmetry of functions- Function symmetry introduction
- Function symmetry introduction
- Even and odd functions: Graphs
- Even and odd functions: Tables
- Even and odd functions: Graphs and tables
- Even and odd functions: Equations
- Even and odd functions: Find the mistake
- Even & odd functions: Equations
- Symmetry of polynomials
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Function symmetry introduction
Learn what even and odd functions are, and how to recognize them in graphs.
What you will learn in this lesson
A shape has reflective symmetry if it remains unchanged after a reflection across a line.
For example, the pentagon above has reflective symmetry.
Notice how line
This idea of reflective symmetry can be applied to the shapes of graphs. Let's take a look.
Even functions
A function is said to be an even function if its graph is symmetric with respect to the
For example, the function
Verify this for yourself by dragging the point on the
Check your understanding
An algebraic definition
Algebraically, a function
For example, for the even function below, notice how the
Odd functions
A function is said to be an odd function if its graph is symmetric with respect to the origin.
Visually, this means that you can rotate the figure
Another way to visualize origin symmetry is to imagine a reflection about the
For example, the function
Verify this for yourself by dragging the point on the
Check your understanding
An algebraic definition
Algebraically, a function
For example, for the odd function below, notice how the function's symmetry ensures that
Reflection question
Want to join the conversation?
What is the use of describing a function as "even" or "odd"?(58 votes)
Even and odd functions have properties that can be useful in different contexts. The most basic one is that for an even function, if you know f(x), you know f(-x). Similarly for odd functions, if you know g(x), you know -g(x). Put more plainly, the functions have a symmetry that allows you to find any negative value if you know the positive value, or vice versa.(116 votes)
Can an equation be both even and odd?(20 votes)
The only function which is both even and odd is f(x) = 0, defined for all real numbers. This is just a line which sits on the x-axis. If you count equations which are not a function in terms of y, then x=0 would also be both even and odd, and is just a line on the y-axis.(62 votes)
How can you prove definitively that a function is even or odd (or neither) just by its equation? Is there even a way?(16 votes)
Mona's explanation works very well for polynomials. Two things to keep in mind:
1) Odd functions cannot have a constant term because then the symmetry wouldn't be based on the origin.
2) Functions that are not polynomials or that don't have exponents can still be even or odd. For example, f(x)=cos(x) is an even function.(16 votes)
How can a function be neither even or odd?(10 votes)
Even and odd describe 2 types of symmetry that a function might exhibit.
1) Functions do not have to be symmetrical. So, they would not be even or odd.
2) If a function is even, it has symmetry around the y-axis. What is a function has symmetry around y=5? It would not be even, because the symmetry is not around the Y-axis.
3) Similarly, odd functions have symmetry around the origin. Functions might have symmetry based on some point other than the origin. So, they would not be odd.
Hope this helps.(23 votes)
Let's say the parent function y=x^2 gets translated to the left by 4. So now the equation is y=(x+4)^2. Is it still an even function? It is confusing because now the graph is not symmetric over the y-axis. So does this mean it is an odd function now? Or is it neither?(8 votes)- Even function are strictly symmetrical about the y axis, so it's neither.(7 votes)
- In the first question, it asks to select all the even functions. Wouldn't the final option not be correct because it isn't a function?(1 vote)
- Why do you think it isn't a function? Remember, a function is just a set of ordered pairs where each input (x-value) has exactly one output (y-value). The final graph meets that criteria. Functions do not need to a continuous line. They don't even need to be lines, they can just be a collections of points.(17 votes)
- Can a function be neither even nor odd? i think the answer is yes im not quite sure(5 votes)
Yes. There are functions which are neither even nor odd, as they don't exhibit any sort of symmetry. One simple example is f(x) = e^x.(9 votes)
How about symmetry with respect to x-axis only? Is it a thing?
Why did we define an even function to be symmetric with respect to y-axis and not the x one?(5 votes)- Remember the vertical line test? A curve cannot be a function when a vertical line interesects it more than once.
And a curve that is symmetrical around the x-axis will always fail the vertical line test (unless that function is f(x) = 0). So, a function can never be symmetrical around the x-axis.
Just remember:
symmetry around x-axis ≠ function
To answer your second question, "even" and "odd" functions are named for the exponent in this power function:
f(x) = xⁿ
- if n is an even integer, then f(x) is an "even" function
- if n is an odd integer, then f(x) is an "odd" function
Hope this helps!(8 votes)
What is the name for a function that is neither even nor odd?(5 votes)- There is no such terminology, it's just that a function that does not exhibit both the symmetry i.e. even or odd.(6 votes)
Shouldn't the algebraic definition of an odd function be f(x)=-f(-x)? That's what was written in the former video around 2m22s. Unless f(-x)=-f(x) is another way of writing f(x)=-f(-x)? I'm confused.(4 votes)
f(-x) = -f(x) is equivalent to f(x) = -f(-x). Multiplying both sides of one of the equations by -1 will give the other equation.(5 votes)
