Use the function  
f
(
x
)
=
x
x
−
1
  to find the following expressions. Enter your answer as a simplified expression.

f
(
6
x
)
=
  
Correct 
   
6
f
(
x
)
=
  
Correct 
   
f
(
−
x
)
=
  
Correct 
   
f
(
x
−
6
)
=
  
Correct 
   
f
(
x
)
−
6
=
  
  
f
(
x
2
)
=
  


We are given the function \( f(x) = \frac{x}{x-1} \). Now, let's compute the required expressions one by one.

1. **\( f(6x) \)**:
   \[
   f(6x) = \frac{6x}{6x - 1}
   \]
   This is the simplified form.

2. **\( 6f(x) \)**:
   \[
   6f(x) = 6 \times \frac{x}{x-1} = \frac{6x}{x-1}
   \]
   This is the simplified form.

3. **\( f(-x) \)**:
   \[
   f(-x) = \frac{-x}{-x-1} = \frac{-x}{-(x+1)} = \frac{x}{x+1}
   \]
   This is the simplified form.

4. **\( f(x-6) \)**:
   \[
   f(x-6) = \frac{x-6}{(x-6)-1} = \frac{x-6}{x-7}
   \]
   This is the simplified form.

5. **\( f(x) - 6 \)**:
   \[
   f(x) - 6 = \frac{x}{x-1} - 6
   \]
   To simplify, find a common denominator:
   \[
   f(x) - 6 = \frac{x}{x-1} - \frac{6(x-1)}{x-1} = \frac{x - 6(x-1)}{x-1} = \frac{x - 6x + 6}{x-1} = \frac{-5x + 6}{x-1}
   \]
   This is the simplified form.

6. **\( f(x^2) \)**:
   \[
   f(x^2) = \frac{x^2}{x^2-1}
   \]
   This is the simplified form.

---

Would you like further details or clarifications on any of these steps?

Here are 5 additional questions related to function manipulation:

1. What is the domain of \( f(x) = \frac{x}{x-1} \)?
2. How does the function \( f(x) \) behave as \( x \to 1 \)?
3. Can you find \( f'(x) \), the derivative of \( f(x) = \frac{x}{x-1} \)?
4. How would you express \( f^{-1}(x) \), the inverse of \( f(x) \)?
5. What happens to \( f(x) \) as \( x \to \infty \)?

**Tip**: Always check the domain of a function before performing transformations or evaluating expressions.

Use the function f(x) = x / (x - 1) to find the following expressions. Enter your answer as a simplified expression.
f(6x) =
Correct
6f(x) =
Correct
f(-x) =
Correct
f(x - 6) =
Correct
f(x) - 6 =
f(x^2) =

Explore how to simplify function expressions using f(x) = x / (x - 1). This includes finding f(6x), 6f(x), f(-x), f(x - 6), and more. Learn step-by-step solutions for rational functions. Suitable for students in Grades 9-11.

Math Problem Statement

Use the function
f ( x )

f ( 6 x )

6 f ( x )

f ( − x )

f ( x − 6 )

f ( x ) − 6

f ( x 2 )

Solution

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