Domains, inverses and composition · Domínios, inversos e composição
| English | Português |
|---|---|
| domain/dəˈmeɪn/ | domínio |
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? · Qual descrição define corretamente o domínio?
The set of allowed inputs to a function. · O conjunto de entradas permitidas para uma função.
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 · Domínios, inversos e composição
State the domain and range · Indique o domínio e a imagem
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
For f(x)=2x+3, find f inverse(11). · Para f(x)=2x+3, encontre f⁻¹(11).
Solve 2x+3=11, giving x=4. · Resolva 2x+3=11, resultando em 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). · Para f(x)=2x+3 e g(x)=x², encontre fg(4).
Apply g first: g(4)=16; then f(16)=2×16+3=35. · Aplique g primeiro: g(4)=16; depois f(16)=2×16+3=35.
Every quadratic function on all real numbers has an inverse function. · Toda função quadrática definida para todos os números reais possui uma função inversa.
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. · Elevar ao quadrado pode introduzir soluções estranhas. Restringir o domínio de uma parábola é essencial antes de afirmar a existência de um inverso. Uma translação horizontal dentro de f tem sinal oposto ao movimento do gráfico.
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). · Para f(x)=√(x-2), encontre f(11).
Substitute into the square root: √(11-2)=√9=3. · Substitua na raiz quadrada: √(11-2)=√9=3.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
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.