Composition and Inverse Relationships

Develop composition order, inverse relation, one-to-one check.

Lesson 3 of 43 parts Practice Flashcards

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Composition Order

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

The expression f(g(x)) uses g first and feeds its output into f. The result is defined only where both stages accept their inputs. The first output must be accepted by the second function. 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

$(f∘g)(x)=f(g(x))$

Worked examples

Try itIf f(x)=3x+5 and g(x)=x+7, find f(g(0)).Show answer

Step 1: Find the inside output: g(0)=7. Step 2: Use it as input: f(7)=3(7)+5. Step 3: The result is 26.

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Composition Order

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

Formulas

$(f∘g)(x)=f(g(x))$

Worked examples

Try itIf f(x)=3x+5 and g(x)=x+7, find f(g(0)).Show answer

Step 1: Find the inside output: g(0)=7. Step 2: Use it as input: f(7)=3(7)+5. Step 3: The result is 26.

Inverse Relation

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

An inverse relation reverses ordered pairs. An inverse function exists when each original output corresponds to exactly one original input over the stated domain. When the original function is one-to-one on its domain. 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

f⁻¹(f(x))=x on the appropriate domain

Worked examples

Try itIf f(x)=3x+5, find f⁻¹(26).Show answer

Step 1: Set 3x+5=26. Step 2: Subtract 5 and divide by 3. Step 3: The inverse output is 7.

One-to-One Check

Objective: Explain one-to-one check, apply the rule, and justify relevant restrictions.

A one-to-one function never repeats an output at distinct inputs; its graph passes a horizontal-line test. Domain restriction can make part of a parabola invertible. A full parabola usually repeats outputs on two sides of its vertex. 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

$f(u)=f(v) ⇒ u=v$

Worked examples

Try itHow many inputs give output 26 under f(x)=3x+5?Show answer

Step 1: The slope 3 is nonzero. Step 2: Solving 3x+5=26 gives x=7. Step 3: Exactly one input produces that output.

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