Which Relation Below Represents A One To One Function

Understanding which relation represents a one-to-one function is a foundational concept in algebra and precalculus. Now, it bridges the gap between simple relations—where inputs can have multiple outputs—and the stricter requirements of invertible functions. To determine if a specific relation fits this classification, one must verify two distinct conditions: the relation must be a function, and it must possess the one-to-one property. This article provides a thorough look to identifying one-to-one functions using ordered pairs, graphs, and algebraic rules Small thing, real impact..

What Defines a One-to-One Function?

Before analyzing specific relations, it is crucial to establish the precise definition. A one-to-one function (often denoted as 1-1 or injective) is a function where distinct elements in the domain map to distinct elements in the range.

Formally, a function $f: A \to B$ is one-to-one if and only if for every $x_1, x_2 \in A$: $ \text{If } f(x_1) = f(x_2), \text{ then } x_1 = x_2. $

Equivalently, the contrapositive is often easier to test: $ \text{If } x_1 \neq x_2, \text{ then } f(x_1) \neq f(x_2). $

In plain English: No two different inputs are allowed to produce the same output. Every $y$-value in the range has exactly one $x$-value partner in the domain.

This differs from a standard function, which only requires that one input cannot have multiple outputs (passing the Vertical Line Test). A one-to-one function adds the restriction that one output cannot come from multiple inputs Surprisingly effective..

The Two-Step Verification Process

When presented with a relation—whether as a set of ordered pairs, a mapping diagram, a graph, or an equation—you must run a two-step diagnostic The details matter here..

Step 1: Is it a Function? (The Vertical Line Test)

First, confirm the relation is actually a function.

• Ordered Pairs/Table: Check that no $x$-value repeats with a different $y$-value.
• Graph: Apply the Vertical Line Test (VLT). If any vertical line intersects the graph more than once, it is not a function, and therefore cannot be a one-to-one function.

Step 2: Is it One-to-One? (The Horizontal Line Test)

If it passes Step 1, check the one-to-one condition Most people skip this — try not to..

• Ordered Pairs/Table: Check that no $y$-value repeats with a different $x$-value.
• Graph: Apply the Horizontal Line Test (HLT). If any horizontal line intersects the graph more than once, the function is not one-to-one.
• Algebraic: Assume $f(a) = f(b)$ and try to prove $a = b$. If you can derive $a = b$ algebraically, it is one-to-one.

Analyzing Relations by Representation

Since the prompt asks "which relation below represents a one-to-one function" without providing specific options, we will analyze the four most common formats you will encounter in textbooks and exams.

1. Sets of Ordered Pairs (or Mapping Diagrams)

This is the most direct way to check. You simply scan the coordinates $(x, y)$.

Example A: $ {(1, 2), (2, 3), (3, 4), (4, 5)} $

• Function Check: $x$-values are ${1, 2, 3, 4}$. All unique. Pass.
• One-to-One Check: $y$-values are ${2, 3, 4, 5}$. All unique. Pass.
• Verdict: One-to-One Function.

Example B: $ {(1, 2), (2, 3), (3, 2), (4, 5)} $

• Function Check: $x$-values ${1, 2, 3, 4}$ are unique. Pass.
• One-to-One Check: $y$-values are ${2, 3, 2, 5}$. The output $2$ appears for inputs $1$ and $3$. Fail.
• Verdict: Function, but NOT One-to-One.

Example C: $ {(1, 2), (1, 3), (2, 4)} $

• Function Check: Input $1$ maps to both $2$ and $3$. Fail.
• Verdict: Not a Function (cannot be one-to-one).

2. Graphical Representation

Visual analysis is often the fastest method. You must apply both line tests.

• Strictly Increasing/Decreasing Graphs: Any graph that is strictly increasing (always going up left-to-right) or strictly decreasing (always going down left-to-right) over its entire domain is automatically a one-to-one function. It passes VLT (function) and HLT (one-to-one).
  • Examples: $f(x) = x$, $f(x) = x^3$, $f(x) = e^x$, $f(x) = \ln(x)$.
• Parabolas (Quadratics): $f(x) = x^2$ (standard domain $\mathbb{R}$).
  • VLT: Pass (it's a function).
  • HLT: Fail. A horizontal line $y=4$ hits $(-2, 4)$ and $(2, 4)$.
  • Note: If the domain is restricted to $x \ge 0$, it becomes one-to-one.
• Circles/Ellipses: $x^2 + y^2 = r^2$.
  • VLT: Fail (vertical line hits top and bottom).
  • Verdict: Not a function.
• Horizontal Lines: $f(x) = c$.
  • VLT: Pass.
  • HLT: Fail (the line is the graph; infinite $x$ map to one $y$).
  • Verdict: Function, NOT One-to-One.

3. Algebraic Equations (Formulas)

When given a formula $y = f(x)$, the algebraic test is the gold standard.

The Algebraic Test:

1. Assume $f(a) = f(b)$ for arbitrary $a, b$ in the domain.
2. Manipulate the equation to see if it forces $a = b$.

Case Study: Linear Functions $f(x) = mx + b$ ($m \neq 0$) $ f(a) = f(b) \implies ma + b = mb + b \implies ma = mb \implies a = b $ Result: All non-horizontal linear functions are one-to-one And it works..

Case Study: Quadratic Function $f(x) = x^2$ (Domain: All Real Numbers) $ f(a) = f(b) \implies a^2 = b^2 \implies a = \pm b $ This does not force $a = b$ (e.g., $a=2, b=-2$). Result: NOT one-to-one That's the part that actually makes a difference..

Case Study: Rational Function $f(x) = \frac{1}{x}$ $ f(a) = f(b) \implies \frac{1}{a} = \frac{1}{b} \implies a = b \quad (\text{assuming } a,b \neq 0) $ Result: One-to-One (on its domain $x \neq 0$).

**Case Study: Absolute Value $f(x

|x|$} $ f(a) = f(b) \implies |a| = |b| \implies a = \pm b $ Since $a = -b$ is a valid solution for distinct inputs (e.g., $a=3, b=-3$ both yield $3$), the implication $a=b$ is not forced. Result: NOT one-to-one (on its standard domain $\mathbb{R}$).

Case Study: Square Root Function $f(x) = \sqrt{x}$ (Domain: $x \ge 0$) $ f(a) = f(b) \implies \sqrt{a} = \sqrt{b} \implies a = b \quad (\text{since } a, b \ge 0) $ The non-negative domain restriction eliminates the $\pm$ ambiguity inherent in squaring. Result: One-to-One Nothing fancy..

Case Study: Cubic Function $f(x) = x^3 - 3x$ $ f(a) = f(b) \implies a^3 - 3a = b^3 - 3b $ $ \implies (a^3 - b^3) - 3(a - b) = 0 $ $ \implies (a - b)(a^2 + ab + b^2 - 3) = 0 $ This yields $a = b$ OR $a^2 + ab + b^2 = 3$. Since the second condition can be satisfied by distinct $a$ and $b$ (e.g., $a=-2, b=1$ both give $f(x)=-2$), $a=b$ is not forced. Result: NOT one-to-one (on $\mathbb{R}$). Note: This confirms the graphical "wiggle" of non-monotonic polynomials fails the Horizontal Line Test.

4. Calculus Shortcut: The Derivative Test (For Continuous Differentiable Functions)

If a function $f$ is continuous and differentiable on an interval $I$, you can determine one-to-one status by analyzing the derivative $f'(x)$.

• If $f'(x) > 0$ for all $x \in I$: $f$ is strictly increasing $\implies$ One-to-One.
• If $f'(x) < 0$ for all $x \in I$: $f$ is strictly decreasing $\implies$ One-to-One.
• If $f'(x)$ changes sign on $I$: $f$ has local extrema $\implies$ NOT One-to-One (fails HLT).

Example: $f(x) = x^3 + 2x$ $f'(x) = 3x^2 + 2$. Since $x^2 \ge 0$, $f'(x) \ge 2 > 0$ for all real $x$. Verdict: Strictly increasing everywhere. One-to-One.

Example: $f(x) = x^3 - 3x$ (from previous case study) $f'(x) = 3x^2 - 3 = 3(x-1)(x+1)$. $f'(x) > 0$ on $(-\infty, -1) \cup (1, \infty)$; $f'(x) < 0$ on $(-1, 1)$. Sign change occurs $\implies$ NOT One-to-One on $\mathbb{R}$. Restriction: On the interval $[1, \infty)$, $f'(x) \ge 0$, so it is one-to-one there And that's really what it comes down to..

5. The Ultimate Implication: Inverse Functions

The classification of one-to-one functions is not merely an academic exercise; it is the prerequisite for inverse functions.

A function $f$ has an inverse function $f^{-1}$ if and only if $f$ is one-to-one.

• If $f$ is not one-to-one, the inverse relation $f^{-1}$ fails the Vertical Line Test (it is not a function).
• If $f$ is one-to-one, the graph of $f^{-1}$ is the reflection of $f$ across the line $y=x$, and the domain/range swap roles perfectly.

This changes depending on context. Keep that in mind.

This is why domain restriction is standard practice for non-one-to-one functions like quadratics ($x^2 \to \sqrt{x}$) and trigonometric functions ($\sin x \to \arcsin x$). We surgically remove the "failing" sections of the domain until the Horizontal Line Test passes, guaranteeing a valid inverse function.

Summary Cheat Sheet

**| Calculus | $f'(x)$ never changes sign on the interval $I$ (i.e. $f'(x)>0;\forall x\in I$ or $f'(x)<0;\forall x\in I$) | $f'(x)$ has a zero where it crosses from positive to negative (or vice‑versa), producing a local extremum that violates the Horizontal Line Test. |

6. Real‑World Implications of One‑to‑One Behavior

In engineering and physics, many models rely on invertible relationships Simple, but easy to overlook..

• Economics: Supply‑demand curves are often required to be injective to guarantee a unique equilibrium price for a given quantity.
  In practice, * Signal processing: A modulation scheme must be one‑to‑one so that the original waveform can be uniquely recovered from its transformed version. * Cryptography: A permutation of a finite set (a bijection) is the backbone of many symmetric‑key algorithms; without injectivity, decryption would be ambiguous.

You'll probably want to bookmark this section Small thing, real impact..

When a function fails the one‑to‑one test, practitioners typically restrict the domain—just as we do with $x^2$ or $\sin x$—to carve out a region where the mapping becomes invertible.

7. Common Pitfalls and How to Avoid Them

8. Mini‑Practice Set

1. Polynomial: Determine if $f(x)=x^4-2x^2$ is one‑to‑one on $(-\infty,0]$ and on $[0,\infty)$.
2. Trigonometric: Show that $\tan x$ is one‑to‑one on $\bigl(-\frac{\pi}{2},\frac{\pi}{2}\bigr)$ but not on $\mathbb{R}$.
3. Exponential: Prove $g(x)=2^x$ is strictly increasing on $\mathbb{R}$ using its derivative.
4. Rational: Decide whether $h(x)=\frac{x^2+1}{x-1}$ is injective on $(1,\infty)$.

Answers are left for the reader to verify using the methods above.

9. Quick Reference Guide

Easier said than done, but still worth knowing.

10. Conclusion

Understanding whether a function is one‑to‑one is more than a classroom exercise; it unlocks