Inverse graphs and restricted branches
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| one-to-one/wʌn tuː wʌn/ | tương ứng một-một |
Which branch must we choose before reversing it?
- A square model can give the same output from two inputs.
- This lesson studies one-to-one 一一对应: A function for which different inputs always give different outputs.
The method
- An inverse reverses the input-output mapping, so the original function must be one-to-one on its stated domain.
- A horizontal line should meet its graph at most once.
- Swap x and y, solve for the new output and retain the correct branch.
- The inverse graph reflects the original in y=x: each point (a,b) becomes (b,a).
- Original domain and range exchange roles.
For f(x)=(x-1)²+2 on x≥1, why is the inverse 1+√(x-2) and not 1-√(x-2)?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Domain and range swap under inversion. Since f took inputs x≥1, f⁻¹ outputs y≥1, forcing the positive root. The x≤1 domain would instead give 1-√(x-2).
A worked case
Let f(x)=(x−1)²+2 for x≥1. Its range is y≥2. From y=(x−1)²+2, the domain selects x−1≥0, so x=1+√(y−2). Thus f inverse(x)=1+√(x−2), with domain x≥2 and range y≥1. The point (3,6) reflects to (6,3). Checking gives f(f inverse(x))=x for x≥2. In the other order, f inverse(f(x))=1+|x−1|=x for x≥1. Choosing the original domain x≤1 instead gives inverse 1−√(x−2), with range y≤1.
Inverse graphs and restricted branches
An inverse reverses the input-output mapping, so the original function must be one-to-one on its stated domain
Use domains to justify inverse and composite function steps.
For the x≥1 branch, find f inverse(6).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The selected inverse is 1+√(6−2)=3.
Where it goes wrong
- The same quadratic formula on all real inputs is not one-to-one.
- Do not choose the plus square root without using the original domain.
- An inverse function is not the reciprocal 1/f(x).
- A reflected graph needs its domain and range labels as well as its formula.
Not true: Every quadratic on all real inputs has an inverse function. A quadratic is many-to-one on all real inputs, so restrict the domain before finding an inverse.
For f(x)=(x−1)²+2, find f(3).
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
f(3)=(3−1)²+2=6.
Every quadratic on all real inputs has an inverse function.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
A quadratic is many-to-one on all real inputs, so restrict the domain before finding an inverse.
Applying it
- On the branch x≤1, output 6 reverses to input −1, while on x≥1 it reverses to 3.
- Both are correct for their own original domains.
- Check both compositions on their actual domains; applying the inverse identity to an input outside the original domain can fail.
For the x≤1 branch, find the inverse output at input 6.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The left-branch inverse is 1−√(6−2)=−1.
Match each inverse-function fact to its statement.
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
An inverse is not the reciprocal 1/f(x); it undoes f, so f(f⁻¹(x))=x on the inverse’s domain.
In your course
- AQA A-level Mathematics (7357), specification section B.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- One-to-one: A function for which different inputs always give different outputs.
- An inverse reverses the input-output mapping, so the original function must be one-to-one on its stated domain.
- Watch for: the same quadratic formula on all real inputs is not one-to-one.