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1. For each of the following relations, indicate whether it is a function. If it is a function, give its range. If it is not a function, explain why. f:R → R such that f(x) = Vx а. b. f:R → R such that f(x) = Vx²

1. For each of the following relations, indicate whether it is a function. If it is a function, give its range. If it is not a function, explain why. f:R → R such that f(x) = Vx а. b. f:R → R such that f(x) = Vx²

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**1.** For each of the following relations, indicate whether it is a **function**. If it is a function, **give its range**. If it is not a function, **explain why**. **a.** \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = \sqrt{x} \) **b.** \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = \sqrt{x^2} \) *Answer box for 1.* --- **2.** For each of the following functions, indicate whether it is **invertible**. If it is invertible, **give its inverse**. If it is not invertible, **explain why**. **a.** \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(x) = x - 2 \) **b.** \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(x) = 2x + 1 \) *Answer box for 2.* --- **3.** Prove that the following function is **not injective (one-to-one)** and **not surjective (onto)**. Hint: find a **counterexample**. \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = |x - 1| \) *Answer box for 3.*
Transcribed Image Text:**1.** For each of the following relations, indicate whether it is a **function**. If it is a function, **give its range**. If it is not a function, **explain why**. **a.** \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = \sqrt{x} \) **b.** \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = \sqrt{x^2} \) *Answer box for 1.* --- **2.** For each of the following functions, indicate whether it is **invertible**. If it is invertible, **give its inverse**. If it is not invertible, **explain why**. **a.** \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(x) = x - 2 \) **b.** \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) such that \( f(x) = 2x + 1 \) *Answer box for 2.* --- **3.** Prove that the following function is **not injective (one-to-one)** and **not surjective (onto)**. Hint: find a **counterexample**. \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \( f(x) = |x - 1| \) *Answer box for 3.*
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