Domains, inverses and composition
| English | 中文 | Pinyin |
|---|---|---|
| domain/dəˈmeɪn/ | 定义域 | dìng yì yù |
Which inputs are allowed?
- A square-root model returns a real output only for some inputs. Its formula alone does not specify a complete function.
- This lesson studies domain 定义域: The set of allowed inputs to a function.
Choose the mathematical structure
- State the domain and range. For an inverse, first ensure the function is one-to-one on its domain. Composition fg means apply g first, then f; the intermediate output must be an allowed input to f.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines domain?
The set of allowed inputs to a function.
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.
For f(x)=√(x-2), x≥2 and the range is y≥0. From y=√(x-2), x=y²+2. Thus f inverse(x)=x²+2 with x≥0. For g(x)=x+3, fg(1)=f(4)=√2.
Domains, inverses and composition
State the domain and range
Compare the model with the worked case and explain one change.
For f(x)=2x+3, find f inverse(11).
Solve 2x+3=11, giving x=4.
Test a tempting shortcut
- Squaring can introduce extraneous solutions. Restricting a parabola's domain is essential before claiming an inverse. A horizontal translation inside f has the opposite sign to the graph's movement.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Every quadratic function on all real numbers has an inverse function. This claim is false. Explain which definition or assumption it violates.
For f(x)=2x+3 and g(x)=x², find fg(4).
Apply g first: g(4)=16; then f(16)=2×16+3=35.
Every quadratic function on all real numbers has an inverse function.
Squaring can introduce extraneous solutions. Restricting a parabola's domain is essential before claiming an inverse. A horizontal translation inside f has the opposite sign to the graph's movement.
Interpret a new situation
- Check f(f inverse(x))=x on the inverse domain. Use a sketch to test whether a horizontal line meets the original graph more than once.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For f(x)=√(x-2), find f(11).
Substitute into the square root: √(11-2)=√9=3.
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
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.
The set of allowed inputs to a function. Choose the relationship, show the method, check its assumptions and interpret the result.