Algebra II · 35 minutes
Suggested grades 10–12 · Unit 3: Combining and reversing functions · Lesson 2 of 2
Inverse Functions as Reversing a Rule
Solve for the original input and distinguish inverse from reciprocal.
Mr. S Teaches · Original starter lesson · HS-ALG2-06 · Grade range is planning guidance.Watch, notice, discuss
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Introduction to Inverse Functions
By The Organic Chemistry Tutor
Review reversing a rule, reflection across y = x and checking both compositions.
Open on YouTube: Introduction to Inverse Functions (opens a new tab) ↗Finding inverses of rational functions | Equations | Algebra 2 | Khan Academy
By Khan Academy
An extension choice beyond a simple linear inverse: retain restrictions while solving a rational inverse rule.
Open on YouTube: Finding inverses of rational functions | Equations | Algebra 2 | Khan Academy (opens a new tab) ↗What does the inverse undo, and how do the original domain and range exchange roles?
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Learning objective
Find a simple inverse rule and state its domain and interpretation.
Before you begin
- Solving equations
- Function notation
Gather your materials
- Paper and pencil
- Graph paper
- Calculator for checking results
Language to know
- inverse function
- A function reversing a one-to-one input-output correspondence.
- one-to-one
- Different inputs have different outputs.
- reciprocal
- One divided by a nonzero number or expression.
Understand the concept
An inverse answers which original input produced an output. For f(x)=3x−7, solving y=3x−7 for x reverses subtraction and multiplication. The notation f⁻¹ does not mean 1/f. A function must assign only one original input to each inverse input; squaring all real numbers fails this because opposite numbers have equal squares.
Teach it step by step
- Solve y=3x−7 for x and rename the inverse input consistently.
- Use a pair from the original table and reverse its coordinates to check the inverse.
- Find the inverse of h(x)=2x³ by solving h(x)=c.
- Contrast inverse and reciprocal values for f at a chosen nonzero output.
Make the thinking visible
Worked example or model response
f⁻¹(y)=(y+7)/3, so f(5)=8 and f⁻¹(8)=5. For h(x)=2x³, h⁻¹(y)=∛(y/2). The reciprocal 1/f(5)=1/8 is not the inverse result 5.
Guided practice
Find the inverse of k(x)=5x+2 and use it to recover the input producing 27.
Independent practice
1. Find the inverse of m(x)=x/4−3. 2. Recover x from 2x³=54. 3. Explain why x² on all real inputs has no inverse function.
Quick assessment
What distinction is lost if f⁻¹(x) is treated as 1/f(x)?
Answer key & assessment criteria
Answers and teaching guidance
- Guided: k⁻¹(y)=(y−2)/5; input 5 produces 27.
- Independent 1: m⁻¹(y)=4(y+3). 2: x=3. 3: For example 2 and−2 both produce 4, so reversal would assign two outputs to input 4.
- Quick check: An inverse reverses a correspondence; a reciprocal divides 1 by an output and performs a different operation.
Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.
Make it work for your learner
More support
Use reversed input-output tables before symbolic inversion.
Extension challenge
Find the inverse of(x+1)/(x−1), carefully stating the excluded input and output.
Verified references · Selected contributions
Standards connections
Subject-content standards
These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.
Contributes to Common Core standard
HSF.BF.B.4.a
Common Core · High school · Functions
Regular high school mathematics standard
Solve an invertible simple rule for its input and write the inverse expression.
Read the official standard ↗This lesson contributes selected skills; broader applications and full standard coverage require further instruction.
Verified 2026-11-04. Selected contribution, not full attainment.