Composition and Inverse Relationships

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

Lesson 4 of 43 parts Practice Flashcards

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Composition Order

Objective: Explain composition order and use it to solve or interpret a problem.

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.

Worked examples

Try itComposition and Inverse Relationships — practice 1.1: If f(x)=4x+3 and g(x)=x+5, find f(g(0)).Show answer

Step 1: Find the inside output: g(0)=5. Step 2: Use it as input: f(5)=4(5)+3. Step 3: The result is 23.

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Composition Order

Objective: Explain composition order and use it to solve or interpret a problem.

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.

Worked examples

Try itComposition and Inverse Relationships — practice 1.1: If f(x)=4x+3 and g(x)=x+5, find f(g(0)).Show answer

Step 1: Find the inside output: g(0)=5. Step 2: Use it as input: f(5)=4(5)+3. Step 3: The result is 23.

Inverse Relation

Objective: Explain inverse relation and use it to solve or interpret a problem.

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.

Worked examples

Try itComposition and Inverse Relationships — practice 2.1: If f(x)=2x+4, find f⁻¹(20).Show answer

Step 1: Set 2x+4=20. Step 2: Subtract 4 and divide by 2. Step 3: The inverse output is 8.

One-to-One Check

Objective: Explain one-to-one check and use it to solve or interpret a problem.

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.

Worked examples

Try itComposition and Inverse Relationships — practice 3.1: How many inputs give output 38 under f(x)=3x+5?Show answer

Step 1: The slope 3 is nonzero. Step 2: Solving 3x+5=38 gives x=11. Step 3: Exactly one input produces that output.

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