Function Notation, Evaluation, and Domain

7.5: Function Notation, Evaluation, and Domain. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 5 of 83 parts Practice Flashcards

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Function inputs

Objective: Evaluate a function at a number or expression.

Prerequisite skills: function notation and real-number arithmetic. A function assigns one output to each allowed input. In f(a), replace every occurrence of the independent variable with a, preserving parentheses when powers or signs occur. Common error: squaring only part of a compound input. Restrictions and interpretation: Inputs must satisfy every original denominator and even-root restriction.

Formulas

$If f(x)=x²+c, then f(a)=a²+c$

Worked examples

Try itIf f(x) = x² + 26, what is f(−23)?Show answer

Replace x by −23: f(−23)=(−23)²+26=529+26=555. The parentheses make the whole negative number the base. Therefore the requested answer is 555.

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Function inputs

Objective: Evaluate a function at a number or expression.

Formulas

$If f(x)=x²+c, then f(a)=a²+c$

Worked examples

Try itIf f(x) = x² + 26, what is f(−23)?Show answer

Replace x by −23: f(−23)=(−23)²+26=529+26=555. The parentheses make the whole negative number the base. Therefore the requested answer is 555.

Domain of a radical

Objective: Determine the allowed inputs for an even-root function.

A real square root requires its entire radicand to be nonnegative. Solve that inequality for the input. Equality is permitted unless another restriction excludes the boundary. Common error: requiring a radicand to be strictly positive when zero is valid. Restrictions and interpretation: Inputs must satisfy every original denominator and even-root restriction.

Formulas

$√(ax+b) real ⇒ ax+b ≥ 0$

Worked examples

Try itWhat is the real domain of f(x) = √(3x − 72)?Show answer

Require 3x−72≥0. Divide by positive 3: x≥24. At the endpoint the radicand is zero, which is allowed. Therefore the requested answer is x ≥ 24.

Combined domain restrictions

Objective: Intersect radical and denominator conditions.

For a quotient involving a square root, impose both the radicand inequality and the nonzero denominator condition. A canceled expression does not restore an excluded original input. Common error: checking only the radicand and ignoring the denominator. Restrictions and interpretation: Inputs must satisfy every original denominator and even-root restriction. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$√(x−a)/(x−b): x≥a and x≠b$

Worked examples

Try itWhat is the real domain of f(x) = √(x − 23)/(x − 26)?Show answer

The radicand requires x−23≥0, hence x≥23. The denominator requires x≠26. Both conditions must hold simultaneously. Therefore the requested answer is x ≥ 23, with x ≠ 26.

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