4. Let f : R –→ R be defined by f(x) = 2x +3. (i) Show that f is 1-1. (ii) Show that ƒ is onto. (iii) Show that f (x) = x² from R to R is not 1-1. (iv) Show that f(x) = x? from R to R is not onto.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Topic Video
Question

Hi,

 

I am struggling with the attached HW problem. Any help is appreciated. Thanks.

### Mathematics Problem: Function Properties

Consider the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 2x + 3 \).

(i) **Show that \( f \) is one-to-one (1-1).**

(ii) **Show that \( f \) is onto.**

(iii) **Show that \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is not one-to-one (1-1).**

(iv) **Show that \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is not onto.**

This problem set asks to explore the properties of two different functions: \( f(x) = 2x + 3 \) and \( f(x) = x^2 \), specifically focusing on their injectivity (one-to-one nature) and surjectivity (onto nature).

To approach these problems, one must:

1. For one-to-one (injectivity), demonstrate that if \( f(x_1) = f(x_2) \) then \( x_1 = x_2 \).
2. For onto (surjectivity), prove that for every \( y \) in the codomain, there exists an \( x \) in the domain such that \( f(x) = y \).

Careful steps and logical arguments must be used to rigorously show these properties for each function in the problems provided.
Transcribed Image Text:### Mathematics Problem: Function Properties Consider the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 2x + 3 \). (i) **Show that \( f \) is one-to-one (1-1).** (ii) **Show that \( f \) is onto.** (iii) **Show that \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is not one-to-one (1-1).** (iv) **Show that \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \) is not onto.** This problem set asks to explore the properties of two different functions: \( f(x) = 2x + 3 \) and \( f(x) = x^2 \), specifically focusing on their injectivity (one-to-one nature) and surjectivity (onto nature). To approach these problems, one must: 1. For one-to-one (injectivity), demonstrate that if \( f(x_1) = f(x_2) \) then \( x_1 = x_2 \). 2. For onto (surjectivity), prove that for every \( y \) in the codomain, there exists an \( x \) in the domain such that \( f(x) = y \). Careful steps and logical arguments must be used to rigorously show these properties for each function in the problems provided.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

Blurred answer
Knowledge Booster
Chain Rule
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,