One-One and Many-One Functions
A function gives one output for each input. A also prevents two different inputs from sharing an output.
One-one vs many-one
- One-one: different inputs give different outputs, like .
- : two inputs can share an output, like , where and both give .
- Horizontal-line test: if no horizontal line cuts the graph more than once on the stated domain, it is one-one.
- Only a one-one function has an inverse function on that domain. Restricting the domain can turn a many-one rule into a one-one function.
On the stated domain: one-one = passes the horizontal-line test = has an inverse function.
Your turnIndependent concept check
The rule is . Decide whether it is one-one when (a) and (b) .
Show worked answer
On all real numbers, opposite inputs such as and give the same output, so the function is many-one. On , the right half of the parabola rises without repeating an output, so it is one-one.
Answer