What is a power function?
A power function is y = a·xⁿ. The base x changes; the exponent n stays fixed. (Compare an exponential function y = 2ˣ, where the exponent changes.)
- a stretches the graph (|a| > 1), squashes it (|a| < 1) or flips it upside down (a < 0).
- n decides the basic shape.
Examples: y = x², y = 3x³, y = x⁻¹, y = x^(1/2), y = 0.5x⁴.
Whole-number powers: even and odd
Even n (2, 4, 6 …)
- U shape, minimum at (0, 0) when a > 0.
- Even function: f(−x) = f(x), symmetric about the y-axis.
- Domain all real numbers; range y ≥ 0 (for a > 0).
- Decreasing for x < 0, increasing for x > 0.
Odd n (1, 3, 5 …)
- S shape (a line when n = 1).
- Odd function: f(−x) = −f(x), symmetric about the origin (half-turn).
- Domain and range all real numbers; always increasing (a > 0).
Bigger n: flatter near 0 (between −1 and 1) and steeper beyond 1. All pass through (1, 1); even ones through (−1, 1), odd ones through (−1, −1).
Negative powers: hyperbolas and asymptotes
x⁻ⁿ = 1/xⁿ. Because 1/0 is not defined, x = 0 is not in the domain.
- y = 1/x (n = −1, odd): two branches in quadrants I and III. Odd function.
- y = 1/x² (n = −2, even): both branches above the x-axis. Even function.
- Asymptotes: the graph gets ever closer to x = 0 (vertical) and y = 0 (horizontal) but never touches them.
- As x grows, y shrinks: this is inverse variation.
Fractional powers: roots and radical functions
A fractional exponent means a root: x^(1/n) = ⁿ√x, and x^(m/n) = (ⁿ√x)ᵐ.
- y = √x = x^(1/2): domain x ≥ 0, range y ≥ 0. Starts at (0, 0), rises slowly.
- y = ∛x = x^(1/3): defined for all x, since ∛(−8) = −2. Odd function.
- Even roots need x ≥ 0; odd roots work for every x.
Inverse link: y = √x is the inverse of y = x² for x ≥ 0. Inverse graphs are mirror images in the line y = x. Likewise y = ∛x is the inverse of y = x³.
Solving: x³ = 64 → x = 64^(1/3) = 4. √x = 5 → x = 25. Always check answers with even roots.
Properties summary and transformations
| n | Domain | Range (a > 0) | Symmetry |
|---|---|---|---|
| even, > 0 | all x | y ≥ 0 | y-axis |
| odd, > 0 | all x | all y | origin |
| −1 | x ≠ 0 | y ≠ 0 | origin |
| −2 | x ≠ 0 | y > 0 | y-axis |
| 1/2 | x ≥ 0 | y ≥ 0 | none |
| 1/3 | all x | all y | origin |
y = a(x − h)ⁿ + k moves the graph h right and k up. Example: y = (x − 2)² + 1 has its lowest point at (2, 1).
Try it: a paper-square experiment
Cut paper squares of side 1, 2, 3 and 4 cm. Count the 1 cm² boxes inside each: 1, 4, 9, 16. Plot side against area: you have drawn y = x². Now predict the number of 1 cm cubes in cubes of side 1–4 (1, 8, 27, 64) and check with the n = 3 setting in the 3D free play.
Key formulas and definitions
- Power function: y = a·xⁿ
- Even function: f(−x) = f(x); odd function: f(−x) = −f(x)
- x⁻ⁿ = 1 / xⁿ (x ≠ 0)
- x^(1/n) = ⁿ√x; x^(m/n) = (ⁿ√x)ᵐ
- Inverse of y = xⁿ (x ≥ 0 if n even) is y = x^(1/n)
- Shift: y = a(x − h)ⁿ + k
Worked examples
1. Is y = x⁴ even, odd or neither? Find y at x = −2 and x = 2.
(−2)⁴ = 16 and 2⁴ = 16. f(−x) = f(x), so it is even (symmetric about the y-axis).
2. Evaluate 27^(2/3).
27^(1/3) = 3, then 3² = 9.
3. State the domain and range of y = 1/x².
x cannot be 0: domain x ≠ 0. 1/x² is always positive: range y > 0.
4. Solve 2x³ = 250.
x³ = 125, so x = ∛125 = 5.
5. The area of a square is A = s². If the side is multiplied by 3, what happens to the area?
A = (3s)² = 9s². The area becomes 9 times.
6. Find the inverse of f(x) = x³ − 1 and check with x = 2.
y = x³ − 1 → x = ∛(y + 1). So f⁻¹(x) = ∛(x + 1). Check: f(2) = 7, f⁻¹(7) = ∛8 = 2. ✓
7. Where do y = x² and y = x³ meet?
x² = x³ → x²(x − 1) = 0 → x = 0 or 1. Points (0, 0) and (1, 1).
Common mistakes
- Thinking x^(−2) is negative. It means 1/x², which is positive.
- Writing x^(1/2) = x/2. It is √x.
- Forgetting that √x needs x ≥ 0 while ∛x works for negative x.
- Saying x³ > x² for every x. For 0 < x < 1, x³ is smaller.