One-to-one function in AP Precalculus

A one-to-one function is a function in which each output value comes from exactly one input value, so if f(a) = f(b), then a = b. In AP Precalculus, being one-to-one is the condition a function needs (on a given domain) to have an inverse function (EK 2.8.A.1).

What is one-to-one function?

A one-to-one function never repeats an output. Every y-value in the range is hit by exactly one x-value. Formally, if f(a) = f(b), then a = b. Graphically, that means the function passes the horizontal line test, because a horizontal line crossing the graph twice would mean two different inputs share the same output.

Why does AP Precalculus care? Inverses. An inverse function reverses the mapping, sending each output b back to its input a (EK 2.8.A.2). That reversal only works if each output points back to a single input. If f(2) = 9 and f(5) = 9, then f⁻¹(9) has no idea whether to return 2 or 5, so the inverse isn't a function. The fix is domain restriction. The CED says a domain "may be restricted in many ways to make the function invertible" (EK 2.8.A.1). That's exactly what happens with f(x) = x², which fails one-to-one on all reals but becomes invertible if you restrict to x ≥ 0.

Why one-to-one function matters in AP® Precalculus

One-to-one is the gatekeeper concept for Topic 2.8 Inverse Functions in Unit 2 (Exponential and Logarithmic Functions). It directly supports LO 2.8.A (determine the input-output pairs of the inverse of a function) and LO 2.8.B (determine the inverse of a function on an invertible domain). Here's the bigger picture for Unit 2. The entire reason logarithmic functions exist is that exponential functions are one-to-one everywhere, so every exponential has an inverse with no domain restriction needed. When you write log_b(x) as the inverse of b^x, you're cashing in on the one-to-one property. The same idea returns in Unit 3, where sine, cosine, and tangent are wildly not one-to-one and have to be restricted to a small piece of their domain before inverse trig functions can be defined.

How one-to-one function connects across the course

Invertible Function (Unit 2)

These two ideas are nearly the same statement seen from different angles. A function is invertible on a domain exactly when it's one-to-one there. One-to-one describes the function's behavior; invertible describes the payoff, that f⁻¹ exists and swaps the (a, b) pairs to (b, a).

Horizontal Line Test (Unit 2)

This is the graphical version of one-to-one. If no horizontal line hits the graph more than once, no output is repeated, so the function is one-to-one. It pairs nicely with the inverse-as-reflection idea in EK 2.8.B.3, since reflecting over y = x turns horizontal lines into vertical ones.

Composite Function (Unit 2)

Composition is how you verify an inverse. EK 2.8.B.1 says f(f⁻¹(x)) = f⁻¹(f(x)) = x on the invertible domain. That identity only holds when f is one-to-one there, because otherwise f⁻¹ isn't even a function you can compose with.

Exponential and Logarithmic Functions (Unit 2)

Exponential functions are always increasing or always decreasing, which makes them automatically one-to-one on their whole domain. That's why logarithms exist as full-fledged inverse functions with no domain restriction, unlike x² or the trig functions you'll restrict later in Unit 3.

Is one-to-one function on the AP® Precalculus exam?

Expect this concept inside inverse-function questions rather than as a standalone vocab check. Multiple-choice stems ask things like what feature a one-to-one function must have (each output mapped from a unique input) or what to do when a function is not one-to-one (restrict the domain to a piece where it is). Modeling-style questions push the same idea in context. For example, given a population function P(t), you might be asked what must be true for P⁻¹(40) to be a function. The answer is that P must be one-to-one on the relevant domain; if the population hit 40 thousand at two different times, P⁻¹(40) wouldn't give a single value, so you'd restrict the domain to one interval. Be ready to check one-to-one from a table (no repeated outputs), a graph (horizontal line test), or an equation (show f(a) = f(b) forces a = b), and to name a valid restricted domain.

One-to-one function vs Function (vertical line test)

The vertical line test checks whether a graph is a function at all, meaning each input has one output. One-to-one is a stricter, second-level check using the horizontal line test, meaning each output also has only one input. Every one-to-one function is a function, but plenty of functions (like y = x² on all reals) are not one-to-one. Memory hook: vertical line test = is it a function, horizontal line test = does it have an inverse function.

Key things to remember about one-to-one function

• A one-to-one function never repeats an output value, so if f(a) = f(b), then a must equal b.

• One-to-one is the condition for invertibility. EK 2.8.A.1 says a function has an inverse on a domain exactly when each output comes from a unique input.

• Graphically, one-to-one means the function passes the horizontal line test, which is different from the vertical line test that checks whether the graph is a function at all.

• If a function isn't one-to-one, you can restrict its domain to an interval where it is, like keeping x ≥ 0 for f(x) = x².

• From a table, check one-to-one by scanning the outputs for repeats; the inverse just swaps each pair (a, b) to (b, a).

• Exponential functions are one-to-one on their entire domain, which is why logarithms exist as their inverses without any restriction.

Frequently asked questions about one-to-one function

What is a one-to-one function in AP Precalculus?

It's a function where every output value corresponds to exactly one input value, so no y-value gets repeated. This is the requirement (from Topic 2.8) for a function to have an inverse function on a given domain.

Is every function one-to-one?

No. Being a function only requires one output per input. One-to-one adds the reverse condition, one input per output. y = x² is a function but not one-to-one, since f(2) = f(-2) = 4.

How is one-to-one different from invertible?

They're two sides of the same coin. One-to-one describes the function's behavior (no repeated outputs), and invertible is the consequence (f⁻¹ exists). On any domain where a function is one-to-one, it's invertible there, per EK 2.8.A.1.

Can a function that isn't one-to-one ever have an inverse?

Yes, but only after you restrict its domain to a piece where it is one-to-one. That's exactly how x² gets the inverse √x (restrict to x ≥ 0) and how the trig functions get inverse trig functions in Unit 3.

Which line test checks if a function is one-to-one?

The horizontal line test. If every horizontal line crosses the graph at most once, the function is one-to-one. The vertical line test is the separate check for whether the graph is a function in the first place.