Title: How to Determine If a Function Is One-to-One – Algebra and Graph Methods

This post explains how to determine whether a function is one-to-one using both algebraic logic and the horizontal line test. This is a core concept in precalculus, especially when dealing with inverse functions.

Key Concept:

A function is one-to-one if:

• No two different values of x yield the same value of y.

• A horizontal line intersects the graph at most once.

Example 1: f(x)=x2f(x) = x^2

This function is not one-to-one:

• f(−2)=4f(-2) = 4 and f(2)=4f(2) = 4 → x1≠x2x_1 \ne x_2, but f(x1)=f(x2)f(x_1) = f(x_2)

• The graph fails the horizontal line test (a horizontal line crosses it twice)

• Final conclusion: Not one-to-one

Example 2: f(x)=2+x+3f(x) = 2 + \sqrt{x + 3}

This function is one-to-one:

• If f(x1)=f(x2)f(x_1) = f(x_2), then x1+3=x2+3\sqrt{x_1 + 3} = \sqrt{x_2 + 3} → x1=x2x_1 = x_2

• No two different x-values will yield the same y-value

• Passes the horizontal line test

• Final conclusion: One-to-one

Simplified Definitions:

• A function is not one-to-one if a horizontal line crosses the graph more than once.

• A function is one-to-one if every unique x-value produces a unique y-value.

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