15. The Function $f(x$\] Has A Domain Of $x \ \textless \ 3$ And A Range $y \geq -5$. What Is The Domain And Range Of $f^{-1}(x$\]?

Understanding the Original Function

The given function $f(x)$ has a domain of $x < 3$ and a range of $y \geq -5$. This means that the function is defined for all values of $x$ less than 3, and the output of the function will always be greater than or equal to -5.

The Concept of Inverse Functions

An inverse function, denoted as $f^{-1}(x)$, is a function that reverses the operation of the original function. In other words, if $f(x)$ maps an input $x$ to an output $y$, then $f^{-1}(x)$ maps the output $y$ back to the input $x$. The inverse function is a one-to-one function, meaning that each output value corresponds to exactly one input value.

Finding the Domain and Range of the Inverse Function

To find the domain and range of the inverse function $f^{-1}(x)$, we need to consider the following:

• The domain of the inverse function is the range of the original function, which is $y \geq -5$.
• The range of the inverse function is the domain of the original function, which is $x < 3$.

Domain of the Inverse Function

The domain of the inverse function $f^{-1}(x)$ is the set of all possible input values for the inverse function. Since the range of the original function is $y \geq -5$, the domain of the inverse function is also $y \geq -5$. However, we need to express this in terms of $x$, so the domain of the inverse function is $x \geq -5$.

Range of the Inverse Function

The range of the inverse function $f^{-1}(x)$ is the set of all possible output values for the inverse function. Since the domain of the original function is $x < 3$, the range of the inverse function is also $x < 3$. However, we need to express this in terms of $y$, so the range of the inverse function is $y < 3$.

Conclusion

In conclusion, the domain and range of the inverse function $f^{-1}(x)$ are $x \geq -5$ and $y < 3$, respectively.

Example

Let's consider an example to illustrate this concept. Suppose we have a function $f(x) = 2x + 1$ with a domain of $x < 3$ and a range of $y \geq -5$. To find the domain and range of the inverse function $f^{-1}(x)$, we can follow the steps outlined above.

• The domain of the inverse function is the range of the original function, which is $y \geq -5$. In terms of $x$, this is $x \geq -5$.
• The range of the inverse function is the domain of the original function, which is $x < 3$. In terms of $y$, this is $y < 3$.

Therefore, the domain and range of the inverse function $f^{-1}(x)$ are $x \geq -5$ and $y < 3$, respectively.

Key Takeaways

• The domain of the inverse function is the range of the original function.
• The range of the inverse function is the domain of the original function.
• The domain and range of the inverse function can be expressed in terms of $x$ and $y$.

Final Thoughts

In conclusion, the domain and range of the inverse function $f^{-1}(x)$ are $x \geq -5$ and $y < 3$, respectively. This is a fundamental concept in mathematics, and it has numerous applications in various fields, including science, engineering, and economics.

Understanding the Original Function

The given function $f(x)$ has a domain of $x < 3$ and a range of $y \geq -5$. This means that the function is defined for all values of $x$ less than 3, and the output of the function will always be greater than or equal to -5.

The Concept of Inverse Functions

An inverse function, denoted as $f^{-1}(x)$, is a function that reverses the operation of the original function. In other words, if $f(x)$ maps an input $x$ to an output $y$, then $f^{-1}(x)$ maps the output $y$ back to the input $x$. The inverse function is a one-to-one function, meaning that each output value corresponds to exactly one input value.

Finding the Domain and Range of the Inverse Function

To find the domain and range of the inverse function $f^{-1}(x)$, we need to consider the following:

• The domain of the inverse function is the range of the original function, which is $y \geq -5$.
• The range of the inverse function is the domain of the original function, which is $x < 3$.

Domain of the Inverse Function

The domain of the inverse function $f^{-1}(x)$ is the set of all possible input values for the inverse function. Since the range of the original function is $y \geq -5$, the domain of the inverse function is also $y \geq -5$. However, we need to express this in terms of $x$, so the domain of the inverse function is $x \geq -5$.

Range of the Inverse Function

The range of the inverse function $f^{-1}(x)$ is the set of all possible output values for the inverse function. Since the domain of the original function is $x < 3$, the range of the inverse function is also $x < 3$. However, we need to express this in terms of $y$, so the range of the inverse function is $y < 3$.

Q&A

Q1: What is the domain of the inverse function $f^{-1}(x)$?

A1: The domain of the inverse function $f^{-1}(x)$ is the range of the original function, which is $y \geq -5$. In terms of $x$, this is $x \geq -5$.

Q2: What is the range of the inverse function $f^{-1}(x)$?

A2: The range of the inverse function $f^{-1}(x)$ is the domain of the original function, which is $x < 3$. In terms of $y$, this is $y < 3$.

Q3: How do we find the domain and range of the inverse function?

A3: To find the domain and range of the inverse function, we need to consider the following:

• The domain of the inverse function is the range of the original function.
• The range of the inverse function is the domain of the original function.

Q4: What is the relationship between the original function and its inverse?

A4: The original function and its inverse are related in such a way that the inverse function reverses the operation of the original function. In other words, if $f(x)$ maps an input $x$ to an output $y$, then $f^{-1}(x)$ maps the output $y$ back to the input $x$.

Q5: What are the key takeaways from this concept?

A5: The key takeaways from this concept are:

• The domain of the inverse function is the range of the original function.
• The range of the inverse function is the domain of the original function.
• The domain and range of the inverse function can be expressed in terms of $x$ and $y$.

Example

Let's consider an example to illustrate this concept. Suppose we have a function $f(x) = 2x + 1$ with a domain of $x < 3$ and a range of $y \geq -5$. To find the domain and range of the inverse function $f^{-1}(x)$, we can follow the steps outlined above.

• The domain of the inverse function is the range of the original function, which is $y \geq -5$. In terms of $x$, this is $x \geq -5$.
• The range of the inverse function is the domain of the original function, which is $x < 3$. In terms of $y$, this is $y < 3$.

Therefore, the domain and range of the inverse function $f^{-1}(x)$ are $x \geq -5$ and $y < 3$, respectively.

Final Thoughts

In conclusion, the domain and range of the inverse function $f^{-1}(x)$ are $x \geq -5$ and $y < 3$, respectively. This is a fundamental concept in mathematics, and it has numerous applications in various fields, including science, engineering, and economics.