Consider The Following Function:${ F(x) = |x-5|, \quad X \geq 5 }$1. Find The Inverse Function { F^{-1} $}$. ${ F^{-1}(x) = \square }$2. State The Domain And Range Of { F $}$. (Enter Your Answers Using

The given function is defined as $f(x) = |x-5|$, where $x \geq 5$. This function represents the absolute value of the difference between $x$ and $5$, and it is only defined for values of $x$ greater than or equal to $5$.

Finding the Inverse Function

To find the inverse function $f^{-1}$, we need to swap the roles of $x$ and $y$ and then solve for $y$. Let's start by writing the given function as $y = |x-5|$.

y = |x-5|

Since $x \geq 5$, we can rewrite the absolute value as a piecewise function:

y = x-5, \quad x \geq 5

Now, we can swap the roles of $x$ and $y$ to get:

x = y-5, \quad x \geq 5

To solve for $y$, we can add $5$ to both sides of the equation:

x+5 = y, \quad x \geq 5

Therefore, the inverse function is given by:

f^{-1}(x) = x+5, \quad x \geq 5

Domain and Range of the Original Function

The domain of the original function $f$ is the set of all values of $x$ for which the function is defined. In this case, the function is only defined for values of $x$ greater than or equal to $5$. Therefore, the domain of $f$ is:

\text{Domain of } f = [5, \infty)

The range of the original function $f$ is the set of all possible values of $f(x)$. Since the function is defined as $f(x) = |x-5|$, the range is the set of all non-negative values greater than or equal to $0$. Therefore, the range of $f$ is:

\text{Range of } f = [0, \infty)

Graphical Representation

The graph of the original function $f$ is a V-shaped graph that opens upwards, with its vertex at $(5, 0)$. The graph of the inverse function $f^{-1}$ is also a V-shaped graph, but it opens to the right and has its vertex at $(0, 5)$.

Conclusion

In this article, we have discussed the given function $f(x) = |x-5|$, where $x \geq 5$. We have found the inverse function $f^{-1}$, which is given by $f^{-1}(x) = x+5$, where $x \geq 5$. We have also determined the domain and range of the original function $f$, which are $[5, \infty)$ and $[0, \infty)$, respectively. Finally, we have provided a graphical representation of the original and inverse functions.

References

• [1] Thomas, G. B. (2014). Calculus and Analytic Geometry. Pearson Education.
• [2] Larson, R. E., & Edwards, B. I. (2013). Calculus: Early Transcendentals. Cengage Learning.

Further Reading

• [1] Khan Academy. (n.d.). Absolute Value Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7f:inverse-functions
• [2] Math Open Reference. (n.d.). Inverse Functions. Retrieved from https://www.mathopenref.com/inversefunctions.html
  Q&A: Understanding the Given Function and its Inverse ===========================================================

In our previous article, we discussed the given function $f(x) = |x-5|$, where $x \geq 5$. We found the inverse function $f^{-1}$, which is given by $f^{-1}(x) = x+5$, where $x \geq 5$. We also determined the domain and range of the original function $f$, which are $[5, \infty)$ and $[0, \infty)$, respectively. In this article, we will answer some frequently asked questions related to the given function and its inverse.

Q1: What is the purpose of finding the inverse function?

A1: The purpose of finding the inverse function is to reverse the operation of the original function. In other words, if we have a function $f(x)$, the inverse function $f^{-1}(x)$ will take the output of $f(x)$ and return the original input.

Q2: How do we know if a function has an inverse?

A2: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, if we have a function $f(x)$, it has an inverse if $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$.

Q3: What is the difference between the original function and its inverse?

A3: The original function $f(x) = |x-5|$ is a V-shaped graph that opens upwards, with its vertex at $(5, 0)$. The inverse function $f^{-1}(x) = x+5$ is also a V-shaped graph, but it opens to the right and has its vertex at $(0, 5)$.

Q4: How do we determine the domain and range of a function?

A4: The domain of a function is the set of all possible input values, while the range is the set of all possible output values. In the case of the original function $f(x) = |x-5|$, the domain is $[5, \infty)$ and the range is $[0, \infty)$.

Q5: Can we have a function with an inverse that is not one-to-one?

A5: No, a function with an inverse must be one-to-one. If a function is not one-to-one, it does not have an inverse.

Q6: How do we find the inverse of a function that is not one-to-one?

A6: If a function is not one-to-one, it does not have an inverse. However, we can find the inverse of a function that is one-to-one by swapping the roles of $x$ and $y$ and then solving for $y$.

Q7: What is the relationship between the original function and its inverse?

A7: The original function $f(x)$ and its inverse $f^{-1}(x)$ are related by the equation $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This means that if we apply the original function to the inverse function, we get the original input, and if we apply the inverse function to the original function, we get the original input.

Q8: Can we have a function with an inverse that is not continuous?

A8: No, a function with an inverse must be continuous. If a function is not continuous, it does not have an inverse.

Q9: How do we determine if a function has an inverse that is continuous?

A9: A function has an inverse that is continuous if it is one-to-one and continuous. In other words, if we have a function $f(x)$, it has an inverse that is continuous if $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$ and $f(x)$ is continuous.

Q10: What is the significance of the inverse function in real-world applications?

A10: The inverse function has many real-world applications, such as modeling population growth, predicting stock prices, and analyzing data. In these applications, the inverse function is used to reverse the operation of the original function and get the original input.

Conclusion

In this article, we have answered some frequently asked questions related to the given function and its inverse. We have discussed the purpose of finding the inverse function, how to determine if a function has an inverse, and the relationship between the original function and its inverse. We have also discussed the significance of the inverse function in real-world applications.

References

• [1] Thomas, G. B. (2014). Calculus and Analytic Geometry. Pearson Education.
• [2] Larson, R. E., & Edwards, B. I. (2013). Calculus: Early Transcendentals. Cengage Learning.

Further Reading

• [1] Khan Academy. (n.d.). Inverse Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f6f4c7f:inverse-functions
• [2] Math Open Reference. (n.d.). Inverse Functions. Retrieved from https://www.mathopenref.com/inversefunctions.html