Inverse graphs and restricted branches
| English | 中文 | Pinyin |
|---|---|---|
| one-to-one/wʌn tuː wʌn/ | 一一对应 | yī yī duì yìng |
A square model can give the same output from two inputs. Which branch must we choose before reversing it?
- A square model can give the same output from two inputs. Which branch must we choose before reversing it?
- This lesson studies one-to-one 一一对应: A function for which different inputs always give different outputs.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines one-to-one?
A function for which different inputs always give different outputs.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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).
The selected inverse is 1+√(6−2)=3.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every quadratic on all real inputs has an inverse function. This claim is false. Explain which definition or assumption it violates.
For f(x)=(x−1)²+2, find f(3).
f(3)=(3−1)²+2=6.
Every quadratic on all real inputs has an inverse function.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For the x≤1 branch, find the inverse output at input 6.
The left-branch inverse is 1−√(6−2)=−1.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A function for which different inputs always give different outputs. Choose the relationship, show the method, check its assumptions and interpret the result.