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
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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![### 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 \( f(x) = x^3 + 7 \):
\[
x_1^3 + 7 = x_2^3 + 7
\]
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcd9c5d72-9ea9-48a4-9d31-06ed7f5c0ae4%2F68dc1801-e9ff-417b-bc0a-02bf8d632dbd%2Fryyxxrp_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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