Let `f:RrarrR` be defined by `f(x)=|x|//x,xne0, f(0)=2`. What is the range of f?

Options

A. {1,2}
B. {1,-1}
C. {-1,1,2}
D. {1}

Explanation

To find the range of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by

\[ f(x) = \begin{cases} \frac{|x|}{x} & \text{if } x \neq 0 \\ 2 & \text{if } x = 0 \end{cases} \]

we will analyze the function for different cases of \( x \).

### Step 1: Analyze \( f(x) \) for \( x > 0 \)

For \( x > 0 \):
- The absolute value function \( |x| \) is equal to \( x \).
- Therefore, we have:
\[ f(x) = \frac{|x|}{x} = \frac{x}{x} = 1 \]

### Step 2: Analyze \( f(x) \) for \( x < 0 \)

For \( x < 0 \):
- The absolute value function \( |x| \) is equal to \( -x \).
- Therefore, we have:
\[ f(x) = \frac{|x|}{x} = \frac{-x}{x} = -1 \]

### Step 3: Analyze \( f(x) \) at \( x = 0 \)

At \( x = 0 \):
- The function is defined as:
\[ f(0) = 2 \]

### Step 4: Determine the range of \( f \)

Now, we can summarize the values of \( f(x) \):
- For \( x > 0 \), \( f(x) = 1 \).
- For \( x < 0 \), \( f(x) = -1 \).
- For \( x = 0 \), \( f(0) = 2 \).

Thus, the possible values of \( f(x) \) are:
- \( -1 \) (for \( x < 0 \))
- \( 1 \) (for \( x > 0 \))
- \( 2 \) (at \( x = 0 \))

### Final Answer

The range of \( f \) is:
\[ \text{Range of } f = \{-1, 1, 2\} \]

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