One-to-One Function
In short: A one-to-one function gives every output value to only one input, so no two different x-values ever land on the same y-value, and that is exactly what allows an inverse.
Algebra 2Functions and GraphsF-BF.B.4a
What is One-to-One Function?
A one-to-one function never repeats an answer. Each member of the range comes from exactly one member of the domain, like a coat-check where every ticket matches exactly one coat.
Formally, \(f\) is one-to-one when \(f(a) = f(b)\) always forces \(a = b\). On a graph, no horizontal line crosses the curve more than once, which is the horizontal line test. Functions that pass this test have inverse functions; those that fail it can often be repaired by restricting the domain.
How to spot it
- A question asks whether a function has an inverse, or whether its inverse is also a function.
- A table of values lists inputs and outputs, and you must decide whether any output repeats.
- A graph rises and falls, and a horizontal line seems able to hit it in two places.
- A domain restriction such as x ≥ 3 is requested to make a parabola invertible.
The method
- For a graph, slide a horizontal line up and down; if it ever meets the curve twice, the rule fails.
- For a table, scan the output column; a repeated output means two inputs share it, so the rule fails.
- For a formula, assume \(f(a) = f(b)\) and simplify; if only \(a = b\) results, the rule passes.
- If it fails, restrict the domain to one side of the repeating part and test again.
One-to-One Function: solved example
Problem
Decide whether \(f(x) = x^2 – 6x + 5\) is one-to-one. If not, restrict the domain so that it is.
Solution
Compute two outputs: \(f(2) = 4 – 12 + 5 = -3\) and \(f(4) = 16 – 24 + 5 = -3\). Different inputs share the output \(-3\), so the function is not one-to-one.
The vertex sits at \(x = 3\), where the curve turns around. Keeping only one side, \(x \ge 3\), removes the repeats and gives a one-to-one function.
Try it
Question 1
Is \(g(x) = x^3 + 2\) one-to-one?
Show the answer
If \(a^3 + 2 = b^3 + 2\), then \(a^3 = b^3\), so \(a = b\). Yes, it is one-to-one.
Question 2
A table gives the pairs (1, 7), (2, 3), (3, 7), (4, 9). Is the relation one-to-one?
Show the answer
No. The inputs 1 and 3 both produce the output 7.
One-to-One Function vs. Function
Every function passes the vertical line test: one input gives one output. Being one-to-one asks for more, namely that one output also comes from only one input, which the horizontal line test checks. So a parabola is a function but not one-to-one over all real numbers.
Keep going
Lessons
Worksheets and quizzes
- Composition and Inverse Functions – Worksheet
- Composition and Inverse Functions – Quiz
- Inverse Functions – Worksheet
- Inverse Functions – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Functions and Graphs