Even and Odd Functions Worksheet- Practice Problems
Even and Odd Functions Worksheets: What You Actually Need
If you're hunting for even and odd functions practice problems, you probably need one of two things: homework help or exam prep. Either way, you're in the right place. This isn't a textbook summary with fluff. It's the actual working material.
Quick Refresher: Even vs. Odd Functions
Before the worksheets, let's nail down the definitions. You can't solve problems you don't understand.
Even Functions
A function is even if f(-x) = f(x) for every x in the domain. The graph is symmetric about the y-axis. Think of it like a mirror reflection vertically.
Odd Functions
A function is odd if f(-x) = -f(x) for every x in the domain. The graph has origin symmetry. Rotate it 180 degrees and it looks the same.
Neither
Most functions fall into this category. They don't satisfy either condition. That's fine. Not everything has to be special.
| Property | Even Functions | Odd Functions |
|---|---|---|
| Definition | f(-x) = f(x) | f(-x) = -f(x) |
| Graph Symmetry | Y-axis symmetry | Origin symmetry |
| Example | f(x) = x² | f(x) = x³ |
| Domain | Unrestricted | Unrestricted |
Practice Problems: Identify the Function Type
Test yourself. Determine whether each function is even, odd, or neither.
Problem Set 1
- Problem 1: f(x) = 3x⁴ - 2x² + 5
- Problem 2: f(x) = 2x³ - x
- Problem 3: f(x) = x² + x
- Problem 4: f(x) = |x|
- Problem 5: f(x) = 5x
Solutions
- Problem 1: Even — Only even powers of x appear. Replace x with -x and you get the same expression.
- Problem 2: Odd — Only odd powers of x. f(-x) = -f(x) checks out.
- Problem 3: Neither — x² is even, x is odd. They don't cancel out. You get f(-x) = x² - x, which isn't f(x) or -f(x).
- Problem 4: Even — |x| = |-x|. Symmetry about the y-axis is obvious.
- Problem 5: Odd — Linear functions with no constant term are odd. f(-x) = -5x = -f(x).
Harder Problems: Verify Algebraically
For these, show your work. Plug in -x and simplify.
Problem Set 2
- Problem 6: f(x) = (x² - 4)/(x + 2)
- Problem 7: f(x) = x⁵ - 4x³ + 2x
- Problem 8: f(x) = √x
- Problem 9: f(x) = cos(x)
- Problem 10: f(x) = sin(x)
Solutions
Problem 6: Simplify first: (x² - 4)/(x + 2) = x - 2 (for x ≠ -2). This is a linear function with no constant term. Odd.
Problem 7: All terms have odd exponents. Factor out x: f(x) = x(x⁴ - 4x² + 2). Since all exponents are even inside the parentheses, f(-x) = -f(x). Odd.
Problem 8: √x is only defined for x ≥ 0. The domain isn't symmetric, so it can't be even or odd. Neither.
Problem 9: cos(-x) = cos(x). This is the definition of an even function. Even.
Problem 10: sin(-x) = -sin(x). This is the definition of an odd function. Odd.
Common Mistakes to Avoid
- Forgetting to simplify first. Some problems require algebra before you can test the condition. Don't jump straight to substitution.
- Assuming all polynomials are even or odd. Only pure even-power polynomials are even. Only pure odd-power polynomials are odd. Mix them and you get neither.
- Ignoring the domain. If the domain isn't symmetric about zero, the function can't be even or odd. Period.
- Arithmetic errors. When you substitute -x, make sure you handle exponents correctly. (-x)² = x², but (-x)³ = -x³.
How to Check Your Work
- Replace every x with -x in the original function.
- Simplify the expression completely.
- Compare the result to your original function and its negative.
- If it matches the original → even. If it matches the negative → odd. If neither → neither.
Quick Reference Cheat Sheet
| Function Type | Test | Examples |
|---|---|---|
| Even | f(-x) = f(x) | x², x⁴, cos(x), |x|, x²sin²(x) |
| Odd | f(-x) = -f(x) | x³, x⁵, sin(x), 1/x, x/(x²+1) |
| Neither | Neither condition holds | x² + x, x + 2, eˣ, √x |
That's the worksheet material. Work through the problems, check your solutions, and move on. Math doesn't require cheerleading. It requires practice.