Draw the graph of function. `y=(1)/(|x|)`

Answer

To draw the graph of the function \( y = \frac{1}{|x|} \), we will analyze the function for different values of \( x \) and sketch the graph accordingly. Here are the steps involved:

### Step 1: Understand the Function
The function \( y = \frac{1}{|x|} \) is defined for all \( x \) except \( x = 0 \) because division by zero is undefined. The absolute value \( |x| \) ensures that the output \( y \) is always positive for \( x \neq 0 \).

### Step 2: Analyze for \( x > 0 \)
For \( x > 0 \), the modulus function does not change the value of \( x \):
\[ y = \frac{1}{|x|} = \frac{1}{x} \]
As \( x \) approaches 0 from the right, \( y \) approaches infinity. As \( x \) increases, \( y \) decreases towards 0.

### Step 3: Analyze for \( x < 0 \)
For \( x < 0 \), the modulus function changes the sign of \( x \):
\[ y = \frac{1}{|x|} = \frac{1}{-x} \]
As \( x \) approaches 0 from the left, \( y \) also approaches infinity. As \( x \) becomes more negative, \( y \) decreases towards 0.

### Step 4: Identify Key Points
- When \( x \) is very close to 0 (from either side), \( y \) is very large (approaching infinity).
- When \( x = 1 \), \( y = 1 \).
- When \( x = -1 \), \( y = 1 \).
- As \( x \) approaches positive or negative infinity, \( y \) approaches 0.

### Step 5: Sketch the Graph
1. Draw the coordinate axes (x-axis and y-axis).
2. For \( x > 0 \), plot the curve of \( y = \frac{1}{x} \) which approaches the y-axis as \( x \) approaches 0 and approaches the x-axis as \( x \) increases.
3. For \( x < 0 \), plot the curve of \( y = \frac{1}{-x} \) which also approaches the y-axis as \( x \) approaches 0 and approaches the x-axis as \( x \) becomes more negative.
4. Mark the asymptotes at \( x = 0 \) (vertical asymptote) and \( y = 0 \) (horizontal asymptote).

### Final Graph
The final graph will consist of two branches:
- One in the first quadrant for \( x > 0 \) (decreasing from infinity to 0).
- One in the second quadrant for \( x < 0 \) (decreasing from infinity to 0).