To solve the equation for \( y \) given \( f(x) = \frac{x + 1}{x - 5} \), we need to express \( y \) in terms of \( x \).

Given:
\[
f(x) = \frac{x + 1}{x - 5}
\]
We set \( f(x) = y \), so:
\[
y = \frac{x + 1}{x - 5}
\]

Now, solve for \( x \) in terms of \( y \):

1. Multiply both sides by \( x - 5 \) to eliminate the denominator:
\[
y(x - 5) = x + 1
\]

2. Distribute \( y \) on the left side:
\[
yx - 5y = x + 1
\]

3. Move all terms involving \( x \) to one side of the equation:
\[
yx - x = 5y + 1
\]

4. Factor out \( x \) on the left side:
\[
x(y - 1) = 5y + 1
\]

5. Finally, solve for \( x \):
\[
x = \frac{5y + 1}{y - 1}
\]

So, the solution for \( x \) in terms of \( y \) is:
\[
x = \frac{5y + 1}{y - 1}
\]

Would you like further details or have any questions?

---

Here are some related questions you might find helpful:
1. How do you verify that \( x = \frac{5y + 1}{y - 1} \) is correct?
2. What happens if \( y = 1 \) in the equation?
3. Can you express \( f^{-1}(x) \) from the function \( f(x) = \frac{x + 1}{x - 5} \)?
4. How does this function behave as \( x \) approaches 5?
5. What is the domain and range of \( f(x) = \frac{x + 1}{x - 5} \)?

**Tip:** Always check the conditions under which the original function is defined, especially when dealing with denominators.

Learn how to solve the equation f(x) = (x + 1) / (x - 5) for y. This problem involves rational functions and equations, providing a step-by-step solution to express y in terms of x. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation