Function machines and reversing operations · Higher
| English | 中文 | Pinyin |
|---|---|---|
| input/ˈɪnpʊt/ | 输入值 | shū rù zhí |
Which input produced the output?
- A machine doubles a number then adds three. Which starting number would produce eleven?
- This lesson studies input 输入值: The value supplied to a rule before its operations are carried out.
Choose the mathematical structure
- Follow the operations in their stated order. To recover a starting input, undo the final operation first. A table pairs each input with its output. The same starting input must have only one output for the rule to be a function.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines input?
The value supplied to a rule before its operations are carried out.
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.
Input 4 gives 2×4+3=11. To reverse output 11, subtract 3 to get 8 then divide by 2 to get 4. Inputs -1,0,1 give outputs 1,3,5. If a second machine squares its input, passing 4 through the first then the second gives 11²=121; reversing their order gives 2×16+3=35.
Function machines and reversing operations
Follow the operations in their stated order
Compare the model with the worked case and explain one change.
Find the output of double then add 3 for input 4.
2×4+3=11.
Test a tempting shortcut
- Reversing the rule does not mean repeating it. The last forward step is the first reverse step. Different operation orders can produce different outputs.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Reversing double then add three means subtracting three only. This claim is false. Explain which definition or assumption it violates.
Find the input producing output 11.
Undo +3, then undo ×2: (11-3)/2=4.
Reversing double then add three means subtracting three only.
Reversing the rule does not mean repeating it. The last forward step is the first reverse step. Different operation orders can produce different outputs.
Interpret a new situation
- AQA A7 Foundation interprets simple functions as input-output rules. Higher also uses formal inverse and composite function notation in the separate function lesson.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the output for input -1.
2×(-1)+3=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
- 8300 · Higher · 3.2. 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.
The value supplied to a rule before its operations are carried out. Choose the relationship, show the method, check its assumptions and interpret the result.