Math Problem Statement
Solution
Let's break down the problem step by step.
We are given the function:
The problem asks us to:
- (a) Find the domain of
- (b) Find the inverse
- (c) Find the domain of the inverse
Part (a): Find the domain of
The domain of a function refers to the set of values for which the function is defined. For this function, there are two main considerations:
-
The fraction inside the square root must be defined, which means the denominator cannot be zero: Solving this: So, .
-
The expression inside the square root must be non-negative (since we're dealing with real numbers): Let's examine when this fraction is non-negative:
- The numerator, , is positive when , which is true for all since is always less than 2.
- The denominator, , is positive when , i.e., , and negative when , i.e., .
Thus, the fraction is non-negative for and negative for .
Conclusion: The domain of is , or .
Part (b): Find the inverse
To find the inverse, we first rewrite the equation for :
Now square both sides to get rid of the square root:
Now, solve for :
- Multiply both sides by :
- Distribute on the left-hand side:
- Bring all terms involving to one side:
- Factor out on the left-hand side:
- Solve for :
- Take the natural logarithm of both sides:
- Solve for :
Thus, the inverse function is:
Part (c): Find the domain of the inverse
The domain of the inverse is the range of the original function. From the form of , we see that:
- The square root ensures that .
- As , .
- As , .
Thus, the range of is , meaning the domain of the inverse function is .
Would you like more detailed steps or further clarification? Here are some related questions for deeper understanding:
- How do you identify when a fraction is non-negative?
- What does the range of a function tell you about its inverse?
- How can we find the domain of more complex expressions like ?
- Why do we take the logarithm when solving for the inverse?
- Can the function ever be negative?
Tip: Always check for restrictions on the function (like square roots or denominators) when determining the domain.
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Math Problem Analysis
Mathematical Concepts
Functions
Domain and Range
Inverse Functions
Logarithms
Formulas
f(x) = (2 - e^(-x)) / (e^(-x) - 1)
Inverse: f^(-1)(y) = -ln((2 + y^2) / (y^2 + 1))
Theorems
Square root and non-negative constraints
Natural logarithm properties for inverse functions
Suitable Grade Level
Undergraduate (Calculus/Pre-Calculus)