A function is even when f(−x) = f(x). Its graph is symmetric about the y-axis, like a mirror.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Odd functions
A function is odd when f(−x) = −f(x). Its graph has rotational symmetry about the origin.
Most are neither
f(x) = x² + x is neither even nor odd. Symmetry is a special property, not a default.
End behaviour
End behaviour describes what happens as x runs far left and far right. For a polynomial the leading term settles it alone.
The two features that decide it
An even degree sends both ends the same way; an odd degree sends them opposite ways. A negative leading coefficient flips both.
Continuity
A function is continuous where its graph has no break. Breaks appear at holes, at jumps and at vertical asymptotes.
Even and odd symmetry
A function is even if f(−x) = f(x), giving symmetry about the y-axis, and odd if f(−x) = −f(x), giving symmetry about the origin. Most functions are neither, and testing takes one substitution.
End behaviour
End behaviour describes what happens as x runs to ±∞. For a polynomial it is decided entirely by the leading term; for a rational function by the relative degrees. It fixes the shape of the graph at the extremes before any plotting.
Continuity, informally
A function is continuous on an interval if you can draw it without lifting the pencil. Breaks come from holes, jumps and vertical asymptotes. Calculus depends on continuity throughout, so identifying it early matters.
Read the graph before computing
Symmetry halves the work; end behaviour bounds the answer; discontinuities warn where a formula will fail. Spending a moment on these before calculating routinely saves more time than it costs.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Kofi Classify f(x) = x³ − 4x
Kofi needs the symmetry and the end behaviour.
- Step 1
He substitutes −x: f(−x) = (−x)³ − 4(−x) = −x³ + 4x.
Step 4: Your Turn
Practice makes it stick.
The Test
Problem 1 of 2
f(x) = x⁴ + 2. Is it even, odd or neither? Enter 1 for even, 2 for odd, 3 for neither.
The Ends
Problem 2 of 2
f(x) = −2x⁴. As x runs far right, does the graph rise or fall? Enter 1 for rise, 2 for fall.
Read the Shape
1 of 8
f(x) = x². Even, odd or neither? 1, 2 or 3.
2 of 8
f(x) = x³. Even, odd or neither? 1, 2 or 3.
3 of 8
f(x) = x² + x. Even, odd or neither? 1, 2 or 3.
4 of 8
f(x) = 3x⁵. How many ends go up?
5 of 8
f(x) = x⁶. How many ends go up?
6 of 8
f(x) = 1 ÷ (x − 3). At which x is it discontinuous?
7 of 8
Sort each function by its symmetry.
Tap something to move it.
- Empty
- Empty
8 of 8
f(x) = −x³. As x runs far right, rise or fall? 1 for rise, 2 for fall.
Step 5: Quick Check
Show what you know.
Question 1 of 2
f(x) = x⁴ − x². Even, odd or neither? 1, 2 or 3.
Question 2 of 2
What decides the end behaviour of a polynomial?
What You Learned
- Even means f(−x) = f(x); odd means f(−x) = −f(x); most functions are neither.
- The leading term alone decides a polynomial’s end behaviour.
- Discontinuities appear at holes, jumps and vertical asymptotes.