Draw the graph of following functions.
`f(x)=|x|+|x-1|`

Answer

To draw the graph of the function \( f(x) = |x| + |x - 1| \), we will analyze the function by breaking it down into intervals based on the points where the expressions inside the absolute values change sign.

### Step 1: Identify critical points
The critical points occur when the expressions inside the absolute values are zero:
1. \( |x| = 0 \) at \( x = 0 \)
2. \( |x - 1| = 0 \) at \( x = 1 \)

This gives us the critical points \( x = 0 \) and \( x = 1 \). We will analyze the function in three intervals:
- \( (-\infty, 0) \)
- \( [0, 1] \)
- \( (1, \infty) \)

### Step 2: Analyze the intervals

#### Interval 1: \( x < 0 \)
In this interval, both \( x \) and \( x - 1 \) are negative:
- \( |x| = -x \)
- \( |x - 1| = -(x - 1) = -x + 1 \)

Thus, the function becomes:
\[ f(x) = -x + (-x + 1) = -2x + 1 \]

#### Interval 2: \( 0 \leq x < 1 \)
In this interval, \( x \) is non-negative and \( x - 1 \) is negative:
- \( |x| = x \)
- \( |x - 1| = -(x - 1) = -x + 1 \)

Thus, the function becomes:
\[ f(x) = x + (-x + 1) = 1 \]

#### Interval 3: \( x \geq 1 \)
In this interval, both \( x \) and \( x - 1 \) are non-negative:
- \( |x| = x \)
- \( |x - 1| = x - 1 \)

Thus, the function becomes:
\[ f(x) = x + (x - 1) = 2x - 1 \]

### Step 3: Combine the results
Now we have the piecewise function:
\[ f(x) = \begin{cases} -2x + 1 & \text{if } x < 0 \\ 1 & \text{if } 0 \leq x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases} \]

### Step 4: Plot the graph
1. **For \( x < 0 \)**: The line \( f(x) = -2x + 1 \) has a y-intercept at \( (0, 1) \) and a slope of -2. It will decrease as \( x \) decreases.
2. **For \( 0 \leq x < 1 \)**: The function is constant at \( f(x) = 1 \).
3. **For \( x \geq 1 \)**: The line \( f(x) = 2x - 1 \) has a y-intercept at \( (0, -1) \) and a slope of 2. It will increase as \( x \) increases.

### Step 5: Mark critical points and draw the graph
- At \( x = 0 \), \( f(0) = 1 \).
- At \( x = 1 \), \( f(1) = 1 \).

Now, we can sketch the graph:
- Start from the left, drawing the line \( f(x) = -2x + 1 \) until \( x = 0 \).
- From \( x = 0 \) to \( x = 1 \), draw a horizontal line at \( y = 1 \).
- From \( x = 1 \) onward, draw the line \( f(x) = 2x - 1 \).

### Final Graph
The graph consists of:
- A decreasing line from \( (0, 1) \) to \( (-\infty, 1) \).
- A horizontal line from \( (0, 1) \) to \( (1, 1) \).
- An increasing line from \( (1, 1) \) onward.