What is a radical function?
A radical function has the input x under a root sign. The simplest one is y = √x. (Some books call it an irrational function.)
For each x you get one answer y. For example √9 = 3, √4 = 2, √1 = 1 and √0 = 0. The answers grow, but slowly. From x = 1 to x = 9 the answer goes only from 1 to 3.
Domain and range of y = √x
Domain means all the x values we are allowed to put in. Range means all the y values we can get out.
We cannot take the square root of a negative number (in real numbers). So x must be 0 or more: x ≥ 0. The root sign always gives 0 or a positive number, so y ≥ 0.
To find the domain of a bigger one such as y = √(3x − 6): the inside must not be negative. 3x − 6 ≥ 0 gives x ≥ 2.
The graph is half a parabola
Start with the sideways parabola x = y². If you take only y ≥ 0, you get y = √x. So the graph of y = √x is the top half of that parabola. It is also the mirror picture of y = x² (for x ≥ 0) in the line y = x. That is why √x is called the inverse of x² on x ≥ 0.
The family y = a√(x − p) + q
Each letter does one job:
- a stretches the curve up (a > 1) or squashes it (0 < a < 1). If a is negative the curve flips down.
- p slides the curve right (p > 0) or left (p < 0). Careful: x − 2 moves it right by 2.
- q slides the curve up (q > 0) or down (q < 0).
The start point is (p, q). The domain is x ≥ p. The range is y ≥ q if a > 0, and y ≤ q if a < 0.
If the x inside has a minus sign, like y = √(−x), the curve is flipped left to right and goes to the left. Then the domain is x ≤ 0.
Inverse and solving radical equations
Finding the inverse. Swap x and y, then solve for y. For y = √(x − 1) + 2 this gives x = √(y − 1) + 2, then (x − 2)² = y − 1, so y = (x − 2)² + 1. We must keep x ≥ 2, because the original range was y ≥ 2.
Solving √(x + 2) = x. Square both sides: x + 2 = x². So x² − x − 2 = 0 and x = 2 or x = −1. Now check each in the first equation. x = 2: √4 = 2 fine. x = −1: √1 = 1 but x = −1, not equal. So −1 is a false answer made by squaring. The only answer is x = 2. Always check!
On a graph, the answer is where the curve y = √(x + 2) meets the line y = x.
Try it
Predict, then check. In the 3D, set a = 1, p = 3, q = −2. Before you look, say the start point, the domain and the range. Then check the readout under the graph. At home: drop a coin from your hand and from a chair, and time both with a phone clock. The chair is about 4 times higher, but the fall is only about 2 times longer.
Key formulas and definitions
- y = √x: domain x ≥ 0, range y ≥ 0, start point (0, 0)
- y = a√(x − p) + q: start point (p, q)
- Domain: x ≥ p. Range: y ≥ q (a > 0) or y ≤ q (a < 0)
- Domain of √(mx + n): mx + n ≥ 0
- Inverse of y = √x (x ≥ 0) is y = x² (x ≥ 0)
Worked examples
1. Find the heights of y = 2√x at x = 0, 1, 4, 9.
Put each x in. x = 0: 2 × 0 = 0. x = 1: 2 × 1 = 2. x = 4: 2 × 2 = 4. x = 9: 2 × 3 = 6. The points are (0, 0), (1, 2), (4, 4), (9, 6).
2. Find the start point, domain and range of y = √(x − 3) + 2.
Compare with a√(x − p) + q: a = 1, p = 3, q = 2. Start point (3, 2). Domain x ≥ 3. Range y ≥ 2 because a is positive.
3. Describe y = −√(x + 1) + 3.
a = −1, p = −1, q = 3. Start point (−1, 3). Domain x ≥ −1. Because a is negative, the curve goes down, so the range is y ≤ 3. Check: at x = 0, y = −1 + 3 = 2.
4. Find the domain of y = √(4 − 2x).
The inside must be 0 or more: 4 − 2x ≥ 0, so 4 ≥ 2x, so x ≤ 2. The domain is x ≤ 2 (the curve goes left).
5. Solve √(x + 2) = x.
Square both sides: x + 2 = x². So x² − x − 2 = 0, which is (x − 2)(x + 1) = 0. x = 2 or x = −1. Check x = 2: √4 = 2 ✓. Check x = −1: √1 = 1 ≠ −1 ✗. Answer: x = 2.
6. Find the inverse of y = √(x − 1) + 2 and say where it is defined.
Swap: x = √(y − 1) + 2. Subtract 2: x − 2 = √(y − 1). Square: (x − 2)² = y − 1. So y = (x − 2)² + 1. The original range y ≥ 2 becomes the new domain: x ≥ 2.
Common mistakes
- Moving the wrong way: y = √(x − 2) slides the curve to the RIGHT by 2, not left.
- Saying the domain is all numbers. Under a root the inside must be 0 or more.
- Forgetting to check after squaring. Squaring can add a false answer, so put every answer back in the first equation.
- Thinking √x has both a plus and a minus answer. The root sign gives only the positive one. The minus half is −√x, a different graph.