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All courses Algebra I · F-IF.5 65 of 76
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 \(f(x)=x^2-4x+1\).
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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 polynomial: its terms use nonnegative integer powers of x and real coefficients.

02

Check for algebraic restrictions

\[\begin{aligned} \text{no}~\text{denominators}~\text{containing}~x \\ \text{no}~\text{even}~\text{roots} \\ \text{no}~\text{logarithms} \end{aligned}\]

The expression has no operation that could be undefined for a real input. Squaring, multiplying, subtracting, and adding are defined for every real x.

03

Distinguish mathematical from contextual domain

\[\text{no}~\text{stated}~\text{context}~\text{or}~\text{interval}~\text{restriction}\]

Nothing in the prompt limits x to time, length, or another physical quantity, so no contextual bound replaces the polynomial's natural domain.

04

State the domain

\[\begin{aligned} \text{domain}=(-\text{infinity},\text{infinity}) \\ \text{all}~\text{real}~\text{numbers} \end{aligned}\]

Every real input produces exactly one real output, so the mathematical domain is all real numbers.

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Another way

  1. A quadratic function ax²+bx+c with real coefficients has domain all real numbers unless a restriction is explicitly stated.

!

Common mistake

Do not find the roots and use the interval between them as the domain. Zeros are allowed inputs for a polynomial and do not restrict where it is defined.

Solution walkthrough video