IF ` f :R -{2} to R ` is a function defined by ` f(x) =(x^(2)-4)/(x-2)` , then its range is

Options

A. `R`
B. `R-{2}`
C. `R-{4}`
D. `R-{-2,2}`

Answer

To find the range of the function \( f(x) = \frac{x^2 - 4}{x - 2} \) where \( f: \mathbb{R} - \{2\} \to \mathbb{R} \), we will follow these steps:

### Step 1: Simplify the function
We start with the function:
\[ f(x) = \frac{x^2 - 4}{x - 2} \]
Notice that \( x^2 - 4 \) can be factored as a difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]
Thus, we can rewrite the function as:
\[ f(x) = \frac{(x - 2)(x + 2)}{x - 2} \]
For \( x \neq 2 \), we can cancel \( x - 2 \):
\[ f(x) = x + 2 \]

### Step 2: Determine the values of \( f(x) \)
The simplified function \( f(x) = x + 2 \) is a linear function. The domain of \( f \) is all real numbers except \( x = 2 \).

### Step 3: Find the value of \( f(2) \)
If we substitute \( x = 2 \) into the simplified function:
\[ f(2) = 2 + 2 = 4 \]
However, since \( x = 2 \) is excluded from the domain, \( f(x) \) will never actually equal 4.

### Step 4: Determine the range
Since \( f(x) = x + 2 \) can take any real number value except for 4 (as \( x \) approaches 2), the range of the function is:
\[ \text{Range of } f = \mathbb{R} - \{4\} \]

### Conclusion
Thus, the range of the function \( f(x) = \frac{x^2 - 4}{x - 2} \) is all real numbers except 4.

### Final Answer
The range of the function is \( \mathbb{R} - \{4\} \).

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