Key takeaway: A one-to-one function gives a different output for every different input, so no horizontal line crosses its graph twice, and that property guarantees an inverse function.
Grade 11 – Algebra 2F-BF.B.4aFunctions and Graphs
Definition of One-to-One Function
A one-to-one function never repeats an output. If two inputs are different, their outputs must be different too. Equivalently, whenever two inputs produce the same output, those inputs were really the same number.
This property matters because it lets you run the function backward. Given an output, exactly one input could have produced it, so an inverse function exists. A parabola fails the property: two inputs on opposite sides of the vertex give the same height. Restricting a function to half of its domain is the usual repair.
Formula and notation
The algebraic test, written with two inputs \(a\) and \(b\):
\[f(a)=f(b) \implies a=b.\]
The visual test is the horizontal line test: if some horizontal line meets the graph at two or more points, the function is not one-to-one. A function with this property is also called an injective function.
Key facts
- Every strictly increasing or strictly decreasing function passes the test.
- Even functions such as x² fail unless the domain is restricted.
- Only a one-to-one function has an inverse that is also a function.
- Linear functions with nonzero slope are always one-to-one.
One-to-One Function example
Is \(g(x)=x^2-6x+4\) a one-to-one function? If not, find a pair of inputs that prove it.
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Rewrite in vertex form: \(g(x)=(x-3)^2-5\). The graph is symmetric about \(x=3\).
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Pick two inputs the same distance from 3, such as 1 and 5.
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Compute: \(g(1)=1-6+4=-1\) and \(g(5)=25-30+4=-1\).
Since \(g(1)=g(5)\) with \(1\neq 5\), g is not one-to-one.
Common mistake
Watch out
Amara says every function is one-to-one, because each input has just one output.
Why it is wrong
That is the definition of a function in general. A one-to-one function adds a second rule: each output must come from just one input. The parabola above gives -1 twice, so it is a function but not one-to-one.
Related terms and lessons
Browse the glossary: A–Z list · Algebra 2 vocabulary · Functions and Graphs