Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Inverse functions (non-linear)
Algebra
CCSS HSF.BF.B.4.ACCSS HSF.BF.B.5
Objective and success criteria
Students will: Inverse functions (non-linear).
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Composite and inverse functions; Logarithms
Leads to: Relate a function's graph to its inverse
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Composite and inverse functions; Logarithms). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- f(x) = 5x − 5 and g(x) = x² + 1. Find f(g(1)). → 5Prompt: Work from the inside out: find g first.
- Evaluate log_3(1). → 0Prompt: Ask: 3 to what power gives 1?
- f(x) = 2x − 4. Find the inverse function f⁻¹(x). → (x+4)/2Prompt: Write y = f(x), then swap x and y.
Hook · 5 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Inverse functions (non-linear). Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: f(x) = 3/(x − 5). Find f⁻¹(x).
- Step 1: x = 3/(y − 5) → y − 5 = 3/x
- Step 2: f⁻¹(x) = 3/x + 5
- Answer: 3/x+5
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- f(x) = √(x − 6), x ≥ 6. Find f⁻¹(x). → x^2+6Prompt: Write y = f(x), swap x and y, then make y the subject.
- What is the inverse of f(x) = 5ˣ? (5⁻ˣ / log_5(x) / x/5 / x⁵) → log_5(x)Prompt: Exponential and logarithmic functions undo each other.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- What is the inverse of f(x) = 3ˣ? (x³ / 1/3ˣ / log_3(x) / x/3) → log_3(x)Prompt: Exponential and logarithmic functions undo each other.
- f(x) = (x − 9)/2. Find f⁻¹(x). → 2x+9Prompt: Write y = f(x), swap x and y, then make y the subject.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Composite and inverse functions.
- Use concrete materials: balance scales or a balance drawing.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Move on to: Relate a function's graph to its inverse.
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder