Math Problem Statement

Y=2x+1/x-5 fungsi invers y (y-1)

Solution

To find the inverse function of y=2x+1x5y = \frac{2x + 1}{x - 5}, we proceed with the following steps:

Step 1: Rewrite the Function

The function given is: y=2x+1x5y = \frac{2x + 1}{x - 5}

To find the inverse, we’ll first replace yy with xx and xx with yy: x=2y+1y5x = \frac{2y + 1}{y - 5}

Step 2: Solve for yy in Terms of xx

Now, we want to isolate yy on one side of the equation.

  1. Start by multiplying both sides by y5y - 5 to get rid of the denominator: x(y5)=2y+1x(y - 5) = 2y + 1

  2. Distribute xx on the left side: xy5x=2y+1xy - 5x = 2y + 1

  3. Move terms involving yy to one side and constants to the other: xy2y=5x+1xy - 2y = 5x + 1

  4. Factor out yy from the left side: y(x2)=5x+1y(x - 2) = 5x + 1

  5. Finally, solve for yy by dividing both sides by x2x - 2: y=5x+1x2y = \frac{5x + 1}{x - 2}

Conclusion

The inverse function of y=2x+1x5y = \frac{2x + 1}{x - 5} is: y1=5x+1x2y^{-1} = \frac{5x + 1}{x - 2}

Would you like additional details or have any questions?

Here are some related questions to consider:

  1. How can we verify if two functions are inverses of each other?
  2. What is the domain and range of the inverse function y1=5x+1x2y^{-1} = \frac{5x + 1}{x - 2}?
  3. How does the process change if we have a different type of function, such as a quadratic?
  4. What applications in real life involve finding inverse functions?
  5. How do we handle inverse functions for functions that are not one-to-one?

Tip: Always check that the function is one-to-one before attempting to find its inverse; this ensures that an inverse function exists.

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Math Problem Analysis

Mathematical Concepts

Algebra
Functions
Inverse Functions

Formulas

y = (2x + 1) / (x - 5)
Inverse function: y = (5x + 1) / (x - 2)

Theorems

Inverse Function Theorem

Suitable Grade Level

Grades 10-12