Questions: Find the domain and range of the rational function. 1. f(x)=2/(x+1)

Solution Steps

To find the domain and range of the rational function \( f(x) = \frac{2}{x+1} \):

1. Domain: Identify the values of \( x \) for which the function is defined. The function is undefined where the denominator is zero.
2. Range: Determine the possible values of \( f(x) \). Since the numerator is a constant and the denominator can take any value except zero, the function can take any real value except zero.

The function \( f(x) = \frac{2}{x+1} \) is undefined when the denominator is zero. Setting the denominator equal to zero gives:

\[ x + 1 = 0 \implies x = -1 \]

Thus, the domain of \( f(x) \) is all real numbers except \( -1 \). In interval notation, this is expressed as:

\[ \text{Domain} = (-\infty, -1) \cup (-1, \infty) \]

To find the range, we analyze the behavior of \( f(x) \). The function can take any real value except for zero, as the numerator is a constant (2) and the denominator can approach zero but never actually be zero. Therefore, the range of \( f(x) \) is:

\[ \text{Range} = (-\infty, 0) \cup (0, \infty) \]

Final Answer