### Bijection of Functions**Determine whether each of these functions is a bijection from \( \mathbb{R} \) to \( \mathbb{R} \). Show all the steps to finding the injective, surjective, and bijective properties.**1. **Function \( f(x) = 3x + 4 \)** - **Injective (One-to-one):** - A function is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \). - For \( f(x) = 3x + 4 \): \[ 3x_1 + 4 = 3x_2 + 4 \] \[ 3x_1 = 3x_2 \] \[ x_1 = x_2 \] - Therefore, \( f(x) = 3x + 4 \) is injective. - **Surjective (Onto):** - A function is surjective if for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( f(x) = y \). - For \( f(x) = 3x + 4 \): \[ y = 3x + 4 \] \[ x = \frac{y - 4}{3} \] - Since \( x \) is a real number for any real \( y \), \( f(x) = 3x + 4 \) is surjective. - **Bijection:** - Since \( f(x) = 3x + 4 \) is both injective and surjective, it is bijective.2. **Function \( f(x) = x^3 + 7 \)** - **Injective (One-to-one):** - A function is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \). - For \( f(x) = x^3 + 7 \): \[ x_1^3 + 7 = x_2^3 + 7 \] \

Answered: Determine whether each of these…

Determine whether each of these functions is a bijection from R to R. Show all the step to finding the injective, surjective and bijective. а. f(x) %3D Зx + 4 b. f(x) = x3 + 7

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

### Bijection of Functions

**Determine whether each of these functions is a bijection from \( \mathbb{R} \) to \( \mathbb{R} \). Show all the steps to finding the injective, surjective, and bijective properties.**

1. **Function \( f(x) = 3x + 4 \)**

- **Injective (One-to-one):**
- A function is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).
- For \( f(x) = 3x + 4 \):
\[
3x_1 + 4 = 3x_2 + 4
\]
\[
3x_1 = 3x_2
\]
\[
x_1 = x_2
\]
- Therefore, \( f(x) = 3x + 4 \) is injective.

- **Surjective (Onto):**
- A function is surjective if for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R} \) such that \( f(x) = y \).
- For \( f(x) = 3x + 4 \):
\[
y = 3x + 4
\]
\[
x = \frac{y - 4}{3}
\]
- Since \( x \) is a real number for any real \( y \), \( f(x) = 3x + 4 \) is surjective.

- **Bijection:**
- Since \( f(x) = 3x + 4 \) is both injective and surjective, it is bijective.

2. **Function \( f(x) = x^3 + 7 \)**

- **Injective (One-to-one):**
- A function is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).
- For \( f(x) = x^3 + 7 \):
\[
x_1^3 + 7 = x_2^3 + 7
\]
\

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