Find the domain of f(x)=1/Sqrt[6x-6] . Explain your steps

To find the domain of the function \( f(x) = \frac{1}{\sqrt{6x - 6}} \), we need to consider the restrictions that the square root and the denominator impose on the function.

### Step 1: Non-negativity of the Square Root Argument
The expression inside the square root, \( 6x - 6 \), must be non-negative because the square root of a negative number is not a real number. Therefore, we set up the inequality:
\[
6x - 6 \geq 0
\]

### Step 2: Solve the Inequality
Solve the inequality for \( x \):
\[
6x \geq 6
\]
\[
x \geq 1
\]

### Step 3: Denominator Cannot Be Zero
The denominator \( \sqrt{6x - 6} \) must not be zero, because division by zero is undefined. Therefore, we set up the condition:
\[
\sqrt{6x - 6} \neq 0
\]
This implies:
\[
6x - 6 \neq 0
\]
\[
6x \neq 6
\]
\[
x \neq 1
\]

### Step 4: Combine the Conditions
From Step 2, we found \( x \geq 1 \). However, from Step 3, we found that \( x \) cannot be equal to 1. Thus, the domain is all values of \( x \) greater than 1.

### Final Answer
The domain of \( f(x) = \frac{1}{\sqrt{6x - 6}} \) is:
\[
\boxed{(1, \infty)}
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the domain of \( f(x) = \frac{1}{\sqrt{4x + 2}} \)?
2. How do you find the domain of a function with both a square root and a logarithm?
3. What is the range of \( f(x) = \frac{1}{\sqrt{6x - 6}} \)?
4. How do you determine the domain of a function involving a cube root?
5. Can you find the domain of \( g(x) = \sqrt{x^2 - 4x + 3} \)?

**Tip:** Always start by checking if the function involves square roots or denominators when determining its domain.

Discover how to find the domain of the function f(x) = 1/Sqrt[6x-6]. This problem involves analyzing the square root and denominator restrictions to determine the valid domain. Suitable for advanced high school students or those studying calculus and functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation