Questions: 3. Determine the domain of f(x). Write your answer in interval notation. 4. Determine the range of f(x). Write your answer in interval notation.

Transcript text: 3. Determine the domain of $f(x)$. Write your answer in interval notation. $\qquad$ 4. Determine the range of $f(x)$. Write your answer in interval notation. $\qquad$ (3 pts)

Solution

Solution Steps

To determine the domain of a function $ f(x) $, we need to identify all the values of $ x $ for which the function is defined. This typically involves checking for values that do not cause division by zero, negative values under even roots, or other undefined operations.

Step 1: Understand the Problem

We need to determine the domain and range of the function $ f(x) $. The domain is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (y-values) that the function can produce.

Step 2: Identify the Function

Since the function $ f(x) $ is not explicitly given in the problem, we will assume a general form of a function and discuss the domain and range for common types of functions. For the sake of this example, let's consider a rational function: \[ f(x) = \frac{1}{x-2} \]

Step 3: Determine the Domain

The domain of a function is all the values of $ x $ for which the function is defined. For the rational function $ f(x) = \frac{1}{x-2} $, the function is undefined when the denominator is zero. Therefore, we need to find the values of $ x $ that make the denominator zero: \[ x - 2 = 0 \] \[ x = 2 \]

Thus, the function is undefined at $ x = 2 $. The domain of $ f(x) $ is all real numbers except $ x = 2 $. In interval notation, this is: \[ (-\infty, 2) \cup (2, \infty) \]

Step 4: Determine the Range

The range of a function is all the possible values of $ y $ that the function can take. For the rational function $ f(x) = \frac{1}{x-2} $, we need to consider the behavior of the function as $ x $ approaches different values:

As $ x $ approaches 2 from the left ($ x \to 2^- $), $ f(x) \to -\infty $.
As $ x $ approaches 2 from the right ($ x \to 2^+ $), $ f(x) \to +\infty $.

Since the function can take any real value except for zero (as there is no $ x $ that makes $ f(x) = 0 $), the range of $ f(x) $ is all real numbers except zero. In interval notation, this is: \[ (-\infty, 0) \cup (0, \infty) \]