Math Problem Statement

What is the inverse of g(x) = 1/(x - 1) and is it a function?

Solution

To find the inverse of the function g(x)=1x1g(x) = \frac{1}{x - 1} and determine whether the inverse is a function, let's follow these steps:

Step 1: Find the inverse of g(x)g(x)

  1. Start by writing y=1x1y = \frac{1}{x - 1}.
  2. Swap xx and yy to find the inverse: x=1y1x = \frac{1}{y - 1}
  3. Solve for yy: x(y1)=1x(y - 1) = 1 xyx=1xy - x = 1 xy=x+1xy = x + 1 y=x+1xy = \frac{x + 1}{x}

So, the inverse function is g1(x)=x+1x=1+1xg^{-1}(x) = \frac{x + 1}{x} = 1 + \frac{1}{x}.

Step 2: Check if the inverse is a function

The inverse, g1(x)=1+1xg^{-1}(x) = 1 + \frac{1}{x}, is indeed a function because each input xx produces a unique output yy, except for x=0x = 0, which is outside the domain.

Conclusion

The correct choice from the given options is:

  • 1x+1, Yes\frac{1}{x} + 1, \text{ Yes}

Would you like more details or have any questions?

Related Questions:

  1. How do you determine the domain and range of the inverse function?
  2. What are the conditions for a function to have an inverse?
  3. How can you verify if an inverse function is a function using the horizontal line test?
  4. What are the asymptotes of the function and its inverse?
  5. How do you graph both a function and its inverse on the same coordinate plane?

Tip: To find the inverse of a function, swap xx and yy and solve for the new yy.

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Algebra

Formulas

y = 1/(x - 1)
x = 1/(y - 1)
y = (x + 1)/x = 1 + 1/x

Theorems

The inverse of a function
Solving for the inverse of a function
Horizontal Line Test for determining if the inverse is a function

Suitable Grade Level

Grades 10-12