Domain And Range Calculator

Q: Domain and range of 2x² − 8x + 3?

Domain is all reals; the vertex is at y = 3 − 64/8 = −5, so the range is y ≥ −5.

Q: Why does a reciprocal exclude y = 0?

Because a/(x + b) can be made arbitrarily small but never exactly zero for finite x.

Work out domain and range instantly with clear inputs, formula shown and shareable results.

About Domain And Range Calculator

The domain is every input the function accepts and the range every output it produces. Restrictions come from three sources: no division by zero, no square root of a negative, and no logarithm of a non-positive number.

Formulas

Quadratic range

vertex k = c − b²/(4a); range is y ≥ k when a > 0 and y ≤ k when a < 0

Square root domain

ax + b ≥ 0

Worked example

With the default values below, Domain And Range Calculator returns Domain of All real numbers. Change any input above and every figure here is recalculated in your browser.

Inputs used

Function typeone of Quadratic ax² + bx + c, Square root √(ax + b), Reciprocal a/(x + b), Logarithm log(ax + b): Quadratic ax² + bx + c
Coefficient a: 2
Coefficient b: -8
Constant c: 3

Results

Domain: All real numbers
Range: y ≥ -5
Critical value: -5.00000000
Function type: quadratic

Step by step

1Values used
2Quadratic range
3Square root domain
4Domain= All real numbers
5Range= y ≥ -5
6Critical value= -5.00000000
7Function type= quadratic

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Domain and range of 2x² − 8x + 3?