20. (i) Write the domain and range of the function $$ |x-2| $$ on $$ R $$ (ii) Is this function differentiable on R.lustify your answer. (ii) Is this function bijective? If not how do we restrict the domain and range of this function so that it become bijective.

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Let's analyze the function $$ f(x) = |x - 2| $$ on the set of real numbers $$ \mathbb{R} $$. We'll address each part of the question in detail.

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### **(i) Domain and Range of $$ f(x) = |x - 2| $$ on $$ \mathbb{R} $$**

**Domain:**
The domain of a function consists of all possible input values (values of $$ x $$) for which the function is defined.

- For $$ f(x) = |x - 2| $$, the absolute value function is defined for all real numbers.
  
**$$\boxed{\text{Domain: } \mathbb{R}}$$**

**Range:**
The range of a function consists of all possible output values (values of $$ f(x) $$) the function can produce.

- The absolute value function $$ |x - 2| $$ always yields a non-negative result because absolute values cannot be negative.
- The smallest value $$ f(x) $$ can attain is 0 (when $$ x = 2 $$).
- As $$ x $$ moves away from 2 in either direction, $$ f(x) $$ increases without bound.
  
**$$\boxed{\text{Range: } [0, \infty)}$$**

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### **(ii) Differentiability of $$ f(x) = |x - 2| $$ on $$ \mathbb{R} $$**

A function is differentiable at a point if its derivative exists at that point. For $$ f(x) = |x - 2| $$:

- **For $$ x > 2 $$:**
  $$
  f(x) = x - 2 \quad \Rightarrow \quad f'(x) = 1
  $$
  
- **For $$ x < 2 $$:**
  $$
  f(x) = -(x - 2) = 2 - x \quad \Rightarrow \quad f'(x) = -1
  $$
  
- **At $$ x = 2 $$:**
  - **Left-Hand Derivative ($$ x \to 2^- $$):** $$ -1 $$
  - **Right-Hand Derivative ($$ x \to 2^+ $$):** $$ 1 $$
  
  Since the left-hand derivative and right-hand derivative at $$ x = 2 $$ are not equal, the derivative does not exist at this point.

**$$\boxed{\text{Differentiable on } \mathbb{R} \setminus \{2\}}$$**

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### **(iii) Bijectivity of $$ f(x) = |x - 2| $$ and Restrictions for Bijectivity**

A function is **bijective** if it is both **injective (one-to-one)** and **surjective (onto)**.

- **Injectivity:**
  - $$ f(x) = |x - 2| $$ is **not injective** on $$ \mathbb{R} $$ because different inputs can yield the same output. For example, $$ f(2 + a) = f(2 - a) = |a| $$ for any $$ a > 0 $$.
  
- **Surjectivity:**
  - Assuming the codomain is $$ \mathbb{R} $$, $$ f(x) $$ is **not surjective** because it only produces non-negative outputs ($$ [0, \infty) $$).
  
Since $$ f(x) $$ is neither injective nor surjective on $$ \mathbb{R} $$, it is **not bijective** in its current form.

**To Make $$ f(x) $$ Bijective:**

We can **restrict the domain** of $$ f(x) $$ to ensure both injectivity and surjectivity. There are two primary ways to do this:

1. **Restrict to $$ x \geq 2 $$:**
   - **Domain:** $$ [2, \infty) $$
   - **Function:** $$ f(x) = x - 2 $$
   - **Range:** $$ [0, \infty) $$
   
   This restriction makes $$ f(x) $$ **injective** because each $$ x \geq 2 $$ maps to a unique $$ y \geq 0 $$, and **surjective** onto $$ [0, \infty) $$.
   
2. **Restrict to $$ x \leq 2 $$:**
   - **Domain:** $$ (-\infty, 2] $$
   - **Function:** $$ f(x) = 2 - x $$
   - **Range:** $$ [0, \infty) $$
   
   Similarly, this restriction ensures $$ f(x) $$ is **bijective** by making it both injective and surjective.

**$$\boxed{\text{By limiting the domain to } [2,\,\infty) \text{ or to } (-\infty,\,2], \text{ the function } f(x) = |x - 2| \text{ becomes bijective.}}$$**

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**Summary:**

1. **Domain:** $$ \mathbb{R} $$
2. **Range:** $$ [0, \infty) $$
3. **Differentiable:** Yes, on $$ \mathbb{R} $$ except at $$ x = 2 $$
4. **Bijective:** No on $$ \mathbb{R} $$. To make it bijective, restrict the domain to $$ [2, \infty) $$ or $$ (-\infty, 2] $$, with the corresponding range $$ [0, \infty) $$.
**Domain:** All real numbers ($$ \mathbb{R} $$) **Range:** Non-negative real numbers ($$ [0, \infty) $$) **Differentiable:** Yes, everywhere except at $$ x = 2 $$ **Bijective:** No on $$ \mathbb{R} $$. To make it bijective, restrict the domain to $$ [2, \infty) $$ or $$ (-\infty, 2] $$, with the range $$ [0, \infty) $$.

20. (i) Write the domain and range of the function \( |x-2| \) on \( R \) (ii) Is this function differentiable on R.lustify your answer. (ii) Is this function bijective? If not how do we restrict the domain and range of this function so that it become bijective.

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