One-to-One Functions - Varsity Tutors

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One-to-One Functions

Study Guide

Key Definition

A function $f$ is one-to-one if no two elements in the domain correspond to the same element in the range. This means each $x$ in the domain has a unique image in the range.

Important Notes

  • Use the Horizontal Line Test to determine if a function is one-to-one.
  • A function $f$ has an inverse $f^{-1}$ if and only if it is one-to-one.
  • The domain of $f$ equals the range of $f^{-1}$ and vice versa.
  • Graphs of a function and its inverse are symmetric with respect to the line $y = x$.
  • If $f(x) = f(y)$, then $x = y$ for one-to-one functions.

Mathematical Notation

$f^{-1}$ represents the inverse of a function$y = mx + b$ is the equation of a line$\sqrt{x}$ represents the square root of $x$$\frac{a}{b}$ represents a fractionRemember to use proper notation when solving problems

Why It Works

One-to-one functions ensure that each input corresponds to a unique output, which allows for the existence of inverse functions.

Remember

The Horizontal Line Test: If no horizontal line intersects the graph of the function more than once, the function is one-to-one.

Quick Reference

Property:$f^{-1}(f(x)) = x$ for every $x$ in the domain of $f$

Understanding One-to-One Functions

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

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Beginner Explanation

A one-to-one function assigns each input a unique output; that is, if $x_1 \neq x_2$ then $f(x_1) \neq f(x_2)$.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Which of the following functions defined on all real numbers is one-to-one?

A$f(x) = x^2$B$f(x) = x + 2$C$f(x) = \frac{1}{x}$D$f(x) = |x|$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

A function represents the amount of money saved over time. Is it one-to-one if the savings strictly increase?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Consider the function $f(x) = 2x + 3$. Prove it's one-to-one.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Which condition ensures that $f(x) = x^3$ is one-to-one?

A$f(x) = x^3$ passes the Horizontal Line TestB$f(x) = x^2$ is increasingC$f(x) = |x|$ is decreasingD$f(x) = \sqrt{x}$ is constantCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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