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Relate the domain of a quadratic function to its graph and situation.

Identify the mathematical domain of a quadratic function with no stated restrictions

Problem
Identify the mathematical domain of quadratic function f(x)=x²-4x+1.
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Hint

Check whether the formula has anything that could restrict inputs, like a denominator or square root of a variable.

A quadratic is a polynomial, and polynomials are defined for every real input.

Solution walkthrough

01

Identify the function type

\[f(x)~=~x^2-4x+1\]

This is a quadratic polynomial because it is built from powers of x with real coefficients.

02

Look for domain restrictions

\[\text{no denominator, no square root, no log}\]

There is nothing in the expression that would make some real x-values invalid.

03

Use the polynomial domain rule

\[\text{polynomials are defined for every real }~x\]

Quadratic functions accept any real number as an input.

04

State the domain

\[\text{domain}:~\text{all}~\text{real}~\text{numbers}\]

So the mathematical domain of this quadratic is all real numbers.

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Another way

  1. Test a few different inputs like 0, -5, and 10; the formula still works each time, which matches the all-real-number domain.

!

Common mistake

A common mistake is to assume a parabola must have a limited domain because its graph has a turning point, even though the formula is defined for every real x.