Find the domain and range of the following:
(a). f(x) = 1/√(x − 3)

Let's analyze the function \( f(x) = \frac{1}{\sqrt{x - 3}} \) to determine its domain and range.

### Domain
The domain of a function consists of all the values of \( x \) for which the function is defined. For the function \( f(x) = \frac{1}{\sqrt{x - 3}} \), there are two conditions that must be satisfied for \( f(x) \) to be defined:

1. The expression under the square root, \( x - 3 \), must be non-negative (since the square root of a negative number is not real).
   \[
   x - 3 \geq 0 \implies x \geq 3
   \]
   
2. The denominator cannot be zero, so \( \sqrt{x - 3} \neq 0 \).
   \[
   \sqrt{x - 3} \neq 0 \implies x - 3 \neq 0 \implies x \neq 3
   \]

Combining these two conditions, the domain of \( f(x) \) is \( x > 3 \).

**Domain:** \( x \in (3, \infty) \)

### Range
The range of the function consists of all possible values of \( f(x) \).

1. \( \sqrt{x - 3} \) is defined for \( x > 3 \), and it will take any positive value as \( x \) increases.
2. As \( \sqrt{x - 3} \) becomes larger, \( f(x) = \frac{1}{\sqrt{x - 3}} \) becomes smaller.
3. As \( x \) approaches 3 from the right, \( \sqrt{x - 3} \) approaches 0, making \( f(x

Learn how to find the domain and range of the function f(x) = 1/sqrt(x - 3) with detailed explanations and step-by-step analysis. This problem covers the concepts of functions, square root functions, and their domains and ranges. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation