Does the relation `{(x,y)|y=|x|,x inR}` define a function? Write the range and draw the graph.

Answer

To determine whether the relation \( \{(x,y) | y = |x|, x \in \mathbb{R}\} \) defines a function, we will follow these steps:

### Step 1: Understand the Definition of a Function
A relation is defined as a function if every input (x-value) corresponds to exactly one output (y-value).

### Step 2: Analyze the Given Relation
The relation given is \( y = |x| \). The absolute value function takes any real number \( x \) and returns its non-negative value.

### Step 3: Check for Unique Outputs
For any real number \( x \):
- If \( x \) is positive, \( |x| = x \).
- If \( x \) is negative, \( |x| = -x \).
- If \( x = 0 \), \( |0| = 0 \).

In all cases, each input \( x \) gives exactly one output \( y \). Therefore, the relation satisfies the criteria for being a function.

### Conclusion for Step 3
Yes, the relation \( \{(x,y) | y = |x|, x \in \mathbb{R}\} \) defines a function.

### Step 4: Determine the Range of the Function
The range of a function is the set of all possible output values \( y \). Since \( y = |x| \) can take any non-negative value:
- The minimum value of \( y \) is 0 (when \( x = 0 \)).
- As \( x \) increases or decreases, \( y \) can take any positive value.

Thus, the range of the function is:
\[ \text{Range} = [0, \infty) \]

### Step 5: Draw the Graph of the Function
To graph \( y = |x| \):
1. Plot points for various values of \( x \):
- \( x = -2 \) gives \( y = 2 \)
- \( x = -1 \) gives \( y = 1 \)
- \( x = 0 \) gives \( y = 0 \)
- \( x = 1 \) gives \( y = 1 \)
- \( x = 2 \) gives \( y = 2 \)

2. The graph consists of two straight lines:
- One line for \( x \geq 0 \) (where \( y = x \)).
- Another line for \( x < 0 \) (where \( y = -x \)).

3. The graph will look like a "V" shape, opening upwards, with the vertex at the origin (0,0).

### Final Summary
- The relation defines a function: **Yes**
- The range of the function: **[0, ∞)**
- The graph is a "V" shape opening upwards.