What does the exponent tell us, and what must we still check about the domain?
My prediction and reason
Before displaying the curves, predict which of the five rules accept −1 and 0, and which can produce a negative output.
Evidence and observations
Complete a comparison table for x, x², x³, √x and 1/x at inputs −1, 0, 1 and 2. Mark undefined entries explicitly.
Record each real domain and range; match each curve with its symmetry and increasing or decreasing intervals.
Under this worksheet’s coefficient-one form y = xᵃ, compare x³ and 2x². State the convention you used rather than treating it as universal terminology.
Explain or give a counterexample
Transfer
A curve includes (1, 1), accepts negative inputs, has no zero input and decreases on each side of zero. Which of the five rules fits? Explain why a negative exponent does not mean all outputs are negative.
Exit task
State the real domain, range and parity of √x. Then state whether 1/x is decreasing across its whole domain.
5 min predict allowed inputs; 12 min compare five curves; 8 min build a property table; 12 min paper identification; 3 min exit check.
What does the exponent tell us, and what must we still check about the domain?
Before displaying the curves, predict which of the five rules accept −1 and 0, and which can produce a negative output.
Evidence tasks
- Complete a comparison table for x, x², x³, √x and 1/x at inputs −1, 0, 1 and 2. Mark undefined entries explicitly.
- Record each real domain and range; match each curve with its symmetry and increasing or decreasing intervals.
- Under this worksheet’s coefficient-one form y = xᵃ, compare x³ and 2x². State the convention you used rather than treating it as universal terminology.
Reference reasoning
This worksheet uses “power function” for the coefficient-one form y = xᵃ with fixed exponent a. Under that convention, x³ is a power function while 2x² is a scalar multiple, not the stated form. Broader texts may use the term differently. For the five real functions: x and x³ have domain and range R and are increasing and odd; x² has domain R, range [0, ∞), is even, decreases on (−∞, 0] and increases on [0, ∞); √x has domain and range [0, ∞), is increasing and neither odd nor even; 1/x has domain and range R excluding zero, is odd, and decreases on each side of zero separately.
Which expression fits the coefficient-one power-function form used in this worksheet?
y=x3andy=2x2 - x³ is exactly x raised to a fixed power, with coefficient 1.
- 2x² has coefficient 2, so it does not fit this stated convention. This is a textbook convention, not a claim about every use of the term.
Find the real domain and range of the square function.
- Every real input can be squared, so the domain is R.
- Outputs are nonnegative; every y ≥ 0 is attained at x = √y. The range is [0, ∞).
Transfer and exit task
A curve includes (1, 1), accepts negative inputs, has no zero input and decreases on each side of zero. Which of the five rules fits? Explain why a negative exponent does not mean all outputs are negative.
State the real domain, range and parity of √x. Then state whether 1/x is decreasing across its whole domain.
√x has domain and range [0, ∞), and is neither odd nor even because its domain is not symmetric. 1/x decreases separately on (−∞, 0) and (0, ∞), but not across the whole domain.
- Includes zero in the square-root domain and range and checks domain symmetry.
- Keeps reciprocal monotonicity on two separate intervals.
Questions
- Which rule rejects zero, and which rejects negative inputs?
- All five curves pass through (1, 1); is that enough to identify a curve?
Expected responses
- Students may mistake a negative exponent for a negative value. Compare 1/2 and 1/(−2).
- Students may call √x even because outputs are nonnegative. Revisit its asymmetric domain.
Confusions
- State the coefficient-one convention when classifying 2x².
- Graph windows hide unbounded outputs and excluded points; rely on the formulas for full domains and ranges.
Board plan
- Five rules: x, x², x³, √x, 1/x.
- Compare domain → range → symmetry → monotonic intervals.
- This worksheet: power-function form y = xᵃ with coefficient 1.
