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Let’s break down and solve the problems step-by-step.

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### **2.1 Find the range for each of the following functions:**

#### a) \( f(x) = x^2 + 1, \; x \in \mathbb{R} \)
- This is a quadratic function with a vertical shift of +1.
- **Vertex:** At \(x = 0\), \(f(0) = 1\). The minimum value of the function is 1.
- As \(x \to \infty\) or \(x \to -\infty\), \(f(x) \to \infty\).

**Range:** \( f(x) \geq 1 \), i.e., \([1, \infty)\).

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#### b) \( g(x) = x^2 + 1, \; x \in \mathbb{R}, \; 1 \leq x \leq 3 \)
- We need to find the minimum and maximum values of the function in the interval \(1 \leq x \leq 3\).
- **At \(x = 1\):** \(g(1) = 1^2 + 1 = 2\).
- **At \(x = 3\):** \(g(3) = 3^2 + 1 = 10\).

**Range:** \( g(x) \in [2, 10] \).

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#### c) \( h(x) = x^2 + 1, \; x \in \mathbb{R}, \; x \leq -1 \)
- As \(x \to -\infty\), \(h(x) \to \infty\).
- **At \(x = -1\):** \(h(-1) = (-1)^2 + 1 = 2\).

**Range:** \( h(x) \geq 2 \), i.e., \([2, \infty)\).

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### **2.2 Find \(f(g(x))\) and \(g(f(x))\):**

#### Given:
- \( f(x) = 2x^2 + 1 \)
- \( g(x) = \sqrt{x}, \; x \geq 0 \)

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#### **1) \( f(g(x)) = f(\sqrt{x}) \):**
- \( f(\sqrt{x}) = 2(\sqrt{x})^2 + 1 = 2x + 1 \).

**Simplified Answer:** \( f(g(x)) = 2x + 1 \).

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#### **2) \( g(f(x)) = g(2x^2 + 1) \):**
- \( g(2x^2 + 1) = \sqrt{2x^2 + 1} \).

**Simplified Answer:** \( g(f(x)) = \sqrt{2x^2 + 1} \).

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Do you want further details on any specific part or have questions?

Here are five follow-up questions:
1. What is the concept of function composition, and why is it useful?
2. How do transformations like shifts affect the range of functions?
3. How can we determine ranges for functions with restricted domains?
4. What properties of square root functions are important in \(g(f(x))\) problems?
5. Why does the range of quadratic functions increase as the input grows?

**Tip:** Always check endpoints carefully when evaluating ranges over restricted domains.

2.1. Find the range for each of the following functions: 
a) f(x) = x^2 + 1, x ∈ R
b) g(x) = x^2 + 1, x ∈ R, 1 ≤ x ≤ 3
c) h(x) = x^2 + 1, x ∈ R, x ≤ -1
2.2. Find f(g(x)) and g(f(x)) if: 
f(x) = 2x^2 + 1 
g(x) = √x, x ≥ 0. Simplify the answer as much as possible.

Understand how to find the range for quadratic functions and evaluate composite functions like f(g(x)) and g(f(x)). This problem covers key algebraic concepts such as range determination, function composition, and the impact of restricted domains. Suitable for students in Grades 9-11.

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