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Functions
Distinguish one output for each input from a one-to-one correspondence.
Start with the picture
A function assigns one output to each allowed input; different inputs may share it.
Check with another example
1. If f(x)=2x+1, what is f(3)?
- 6
- 7
- 4
My reason
2. For f(x)=x², both 2 and −2 give 4. Does that stop it being a function?
- Different inputs must give different outputs
- Input 2 has two outputs
- It is a function: each input has one output
My reason
3. Is g(x)=5, defined for every real x, a function?
- Yes
- No, because the output never changes
- It must allow only one input
My reason
Explain in your own words
Why need a function not be one-to-one? Explain with an example.
Answers and reasons
1. 7
Substituting 3 gives 2×3+1=7.
2. It is a function: each input has one output
Yes. Each of the inputs 2 and −2 has exactly one output.
3. Yes
Each input has exactly one output, 5. A function need not change its output.
Under x², 2 and −2 share output 4, but each input has one output. That satisfies the function rule.