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Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Restrict a domain to make a many-to-one function invertible.
- Justify the conclusion "f⁻¹(y)=1+√y for y≥0" using the stated assumptions.
Before you start
Quadratic graphs and inverse functions.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find the inverse of f(x)=(x−1)² when the domain is x≥1.
Why this math matters
Restrict a domain to make a many-to-one function invertible. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The specified original domain is [1,∞).
- An inverse must return one allowed input per output.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Choose a branch before inverting a quadratic
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the inverse of f(x)=(x−1)² when the domain is x≥1.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
y=(x−1)² implies x−1=±√y
Squaring normally loses the sign of the shifted input.
Work through the mathematics
The restriction x≥1 selects x−1=√y
The allowed branch removes the ambiguity.
Check the conclusion
f⁻¹(y)=1+√y for y≥0
The inverse's domain is the original function's range.
The result
f⁻¹(y)=1+√y for y≥0
The inverse's domain is the original function's range.
Common mistakes to catch
- The ± symbol gives a relation, not one inverse function.
- Swap domain and range when identifying the inverse.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What inverse branch would x≤1 select?
Show a hint
The shifted input is nonpositive.
Reveal answer and explanation
1−√y
The opposite restriction selects the negative square root.
Practice 2
Why does the unrestricted quadratic lack an inverse function?
Show a hint
Compare x=0 and x=2.
Reveal answer and explanation
Both map to one
One output could not recover a unique original input.
Take the idea with you
Explain why a sensor with a symmetric response needs extra information to identify its original input.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Read a horizontal scale from the inside expression
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