Functions, Domain, and Range

Develop function rule, domain restrictions, range and graph reading.

Lesson 1 of 43 parts Practice Flashcards

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Function Rule

Objective: Explain function rule, apply the rule, and justify relevant restrictions.

A function assigns exactly one output to every allowed input. Different inputs can share one output without violating the rule. Yes; the restriction is on multiple outputs for one input. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$y=f(x)$

Worked examples

Try itFor f(x)=3x+5, find f(7).Show answer

Step 1: Replace x with 7: f(7)=3(7)+5. Step 2: Multiply: 21+5. Step 3: Add: 26.

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Function Rule

Objective: Explain function rule, apply the rule, and justify relevant restrictions.

Formulas

$y=f(x)$

Worked examples

Try itFor f(x)=3x+5, find f(7).Show answer

Step 1: Replace x with 7: f(7)=3(7)+5. Step 2: Multiply: 21+5. Step 3: Add: 26.

Domain Restrictions

Objective: Explain domain restrictions, apply the rule, and justify relevant restrictions.

The domain includes only inputs for which a rule is defined and which make sense in context. A denominator cannot be zero and an even root needs a nonnegative radicand over the reals. Making the original denominator zero. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$For 1/(x−a): x≠a$

Worked examples

Try itFor f(x)=1/(x−7), which x-value is excluded?Show answer

Step 1: The denominator cannot be zero. Step 2: Solve x−7=0. Step 3: Therefore x=7 is excluded.

Range and Graph Reading

Objective: Explain range and graph reading, apply the rule, and justify relevant restrictions.

The range consists of outputs actually produced by inputs from the stated domain. A parabola opening upward has a least output at its vertex over the real domain. Changing allowed inputs can change available outputs. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$For y=(x−h)²+k: y≥k$

Worked examples

Try itFor y=(x−7)²+5, find the minimum output.Show answer

Step 1: A square is at least zero. Step 2: At x=7, the square is zero. Step 3: The minimum output is 5.

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