Dx ≥ 0, since x must be non-negative to appear under a square root
For a square root, the radicand must be non-negative: 3 − 2x ≥ 0 → −2x ≥ −3 → x ≤ 3/2. The domain is (−∞, 3/2]. Option D is the most common error: it's the radicand (the expression under the radical) that must be non-negative, not x itself. Option B reverses the inequality — dividing by a negative number (−2) flips the direction.
Question 2 Multiple Choice
Why does f(x) = ∛x accept negative inputs like x = −8, while f(x) = √x does not?
ACube roots are defined differently as a matter of mathematical convention
BNegative numbers have real cube roots because cubing preserves sign, while no real number squared gives a negative result
CThe cube root function uses a different branch cut that allows complex inputs
DBoth functions actually accept all real inputs; √(−8) simply gives an imaginary output
The key distinction is what happens to signs under the inverse operation. Squaring always gives a non-negative result: no real number x satisfies x² = −1, so there is no real square root of a negative number. But cubing preserves sign: (−2)³ = −8, so −2 is a valid cube root of −8. More generally, odd powers preserve sign information, making odd-index radicals (cube root, fifth root, etc.) well-defined for all real numbers. Even powers lose sign, restricting even-index radicals to non-negative radicands in the reals.
Question 3 True / False
The function f(x) = √x always returns a non-negative value, even though every positive number has both a positive and a negative square root.
TTrue
FFalse
Answer: True
By convention, the radical symbol √ denotes the principal (non-negative) square root only. √9 = 3, not ±3. This is why the range of f(x) = √x is [0, ∞). When solving equations like x² = 9, you write x = ±3 because you're finding all numbers whose square is 9 — but the function √9 evaluates to 3 only. This distinction matters for graphing: the graph of y = √x is a half-curve starting at the origin, not a full parabola.
Question 4 True / False
The graph of y = √x is the upper half of the parabola y = x², restricted to x ≥ 0.
TTrue
FFalse
Answer: False
The graph of y = √x is the reflection of the right half of y = x² across the line y = x — not the upper half of y = x². These two functions are inverses of each other (for x ≥ 0), and inverse function graphs are reflections across y = x. On y = x², the input runs left-right and output runs up; on y = √x, input runs left-right and output is the square root, rising slowly. The curve starts at (0, 0) and bends upward with decreasing slope, which is the reflected (not the original) parabola.
Question 5 Short Answer
Explain why the domain of f(x) = √(x − 3) is [3, ∞), and describe how you would find the domain of a general transformed radical function g(x) = √(ax + b).
Think about your answer, then reveal below.
Model answer: The square root requires a non-negative radicand. Setting x − 3 ≥ 0 gives x ≥ 3, so the domain is [3, ∞). For g(x) = √(ax + b): set ax + b ≥ 0 and solve for x. If a > 0, the domain is x ≥ −b/a. If a < 0, dividing by a flips the inequality, giving x ≤ −b/a.
The method is always: set the radicand ≥ 0 (for even-index radicals) and solve the resulting inequality. The transformation h in f(x) = √(x − h) shifts the domain's boundary: the left endpoint of the domain moves from 0 to h. This is a direct consequence of the general transformation rule — replacing x with x − h shifts the graph h units to the right, which shifts the domain boundary from 0 to h.