Determining the domain and range for the function ##f^{-1}##

In summary, the domain of a function is the set of all possible input values and the range is the set of all possible output values. To determine the domain and range, you need to identify any restrictions on the input values and analyze the behavior of the function. For a linear function, the domain and range are all real numbers. For a quadratic function, the domain is determined by restrictions on input values and the range is determined by the shape of the graph. It is possible for the domain and range of a function to be the same if there are no restrictions on the input values and the output values can vary infinitely.
  • #1
angela107
35
2
Homework Statement
For this question, let ##f:[0,+∞)->R## is defined by ##f(x) = √x##, and let ##f^{-1}(x)## be its inverse function. What is the domain and range of the function ##f^{-1}(x)##?
Relevant Equations
n/a
The domain and range of this function will be the same.

We can let ##𝑓(𝑥)=\sqrt{x},𝑥≥0##

However, ##𝑦=𝑓(𝑥)≥0##, so the domain and range of ##f## are ##[0,+∞)##

And since ##f## is a function, ##f^{-1}s## domain is the range of ##f## and ##f^{-1}s## range is ##f’s## domain.

In other words, the domain and range of ##f^{-1}:[0,+∞)→[0,+∞)##.

Is my solution correct?
 
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  • #2
Hi. It seems all right.
 
  • #3
The work is correct. The domain and range for this function are the same -- ##[0, \infty)##, so the domain and range for its inverse are also the same sets.
 

Related to Determining the domain and range for the function ##f^{-1}##

1. What is the purpose of determining the domain and range for a function's inverse?

Determining the domain and range for a function's inverse allows us to understand the input and output values that are valid for the inverse function. This information is crucial for solving equations and understanding the behavior of the inverse function.

2. How do you determine the domain for a function's inverse?

The domain for a function's inverse is determined by finding the range of the original function. This means that the domain for the inverse function will be the same as the range for the original function. It is important to note that the domain for the inverse function may be different from the domain of the original function.

3. How do you determine the range for a function's inverse?

The range for a function's inverse is determined by finding the domain of the original function. This means that the range for the inverse function will be the same as the domain for the original function. It is important to note that the range for the inverse function may be different from the range of the original function.

4. Can the domain and range for a function's inverse be the same as the domain and range of the original function?

Yes, it is possible for the domain and range of a function's inverse to be the same as the domain and range of the original function. This will happen when the original function is a one-to-one function, meaning that each input has a unique output and vice versa.

5. How does determining the domain and range for a function's inverse help in solving equations?

Determining the domain and range for a function's inverse allows us to restrict the possible values for the input and output of the inverse function. This can make solving equations involving the inverse function much easier and more accurate.

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