2.8 One-to-One Functions and Their Inverses
Ideas in this lesson
- Decide and justify one-to-one behavior
- Read inverse values and reverse input-output roles
- Verify inverses and track their domains
- Invert affine and odd-power rules
- Invert rational functions with excluded values
- Restrict domains and invert even powers and radicals
- Construct reflected graphs and test complex graphs
- Interpret inverse models in context
- Invert piecewise and composite models
- Prove identity properties and solve functional equations
Decide and justify one-to-one behavior · Worked example
Is f(x)=\(\left|{x - 4}\right|\) one-to-one on ℝ? Give two distinct inputs with equal outputs to support your answer. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: No: f(3)=f(5)=1.
The inputs 3 and 5 differ.
Both are one unit from 4, so both outputs equal 1.
One repeated output at distinct inputs disproves injectivity.
Written-work review:
Both are one unit from 4, so both outputs equal 1.
One repeated output at distinct inputs disproves injectivity.
Written-work review:
- Two distinct inputs in the domain.
- Equal output demonstrated by substitution.
- Correct conclusion distinguishing function from one-to-one function.