Determining the Domain and Range

Oct 13, 2025 by thesweetdesigns.com 90 views

Introduction

In mathematics, functions are used to describe the relationship between two variables. The function $f(x) = 2 + \sqrt{x - 1}$ is a type of function known as a square root function. In this article, we will explore the graph of this function, determine its domain and range, and find the equation for its inverse.

Graphing the Function

To graph the function $f(x) = 2 + \sqrt{x - 1}$ , we need to understand its behavior. The square root function is defined only for non-negative values of $x$ , so the domain of $f(x)$ is $x \geq 1$ . The graph of $f(x)$ will be a curve that starts at $x = 1$ and increases as $x$ increases.

**Domain:** $x \geq 1$
**Range:** $y \geq 2$

The graph of $f(x)$ will be a curve that starts at $x = 1$ and increases as $x$ increases. The minimum value of $f(x)$ is $2$ , which occurs when $x = 1$ . As $x$ increases, $f(x)$ will increase without bound.

Determining the Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. In this case, the domain of $f(x)$ is $x \geq 1$ , since the square root function is defined only for non-negative values of $x$ .

The range of a function is the set of all possible output values for which the function is defined. In this case, the range of $f(x)$ is $y \geq 2$ , since the minimum value of $f(x)$ is $2$ .

Finding the Inverse Function

To find the inverse function of $f(x)$ , we need to swap the roles of $x$ and $y$ and solve for $y$ . Let $y = f(x) = 2 + \sqrt{x - 1}$ . To find the inverse function, we need to solve for $x$ in terms of $y$ .

**Step 1:** Swap the roles of $x$ and $y$
$y = 2 + \sqrt{x - 1}$
Step 2: Subtract 2 from both sides
 $y - 2 = \sqrt{x - 1}$ 
Step 3: Square both sides
 $(y - 2)^2 = x - 1$ 
Step 4: Add 1 to both sides
 $(y - 2)^2 + 1 = x$ 
Step 5: Simplify
 $x = (y - 2)^2 + 1$

The inverse function of $f(x)$ is $f^{-1}(x) = (x - 2)^2 + 1$ .

Graphing the Inverse Function

To graph the inverse function $f^{-1}(x)$ , we need to understand its behavior. The inverse function is a reflection of the original function across the line $y = x$ . Since the original function is a curve that starts at $x = 1$ and increases as $x$ increases, the inverse function will be a curve that starts at $y = 1$ and increases as $y$ increases.

**Domain:** $x \geq 1$
**Range:** $y \geq 1$

The graph of $f^{-1}(x)$ will be a curve that starts at $y = 1$ and increases as $y$ increases. The minimum value of $f^{-1}(x)$ is $1$ , which occurs when $x = 1$ . As $x$ increases, $f^{-1}(x)$ will increase without bound.

Conclusion

Frequently Asked Questions

In this article, we will answer some frequently asked questions about the function $f(x) = 2 + \sqrt{x - 1}$ and its inverse.

Q: What is the domain of the function $f(x) = 2 + \sqrt{x - 1}$ ?

A: The domain of the function $f(x) = 2 + \sqrt{x - 1}$ is $x \geq 1$ , since the square root function is defined only for non-negative values of $x$ .

Q: What is the range of the function $f(x) = 2 + \sqrt{x - 1}$ ?

A: The range of the function $f(x) = 2 + \sqrt{x - 1}$ is $y \geq 2$ , since the minimum value of $f(x)$ is $2$ .

Q: How do I find the inverse function of $f(x) = 2 + \sqrt{x - 1}$ ?

A: To find the inverse function of $f(x) = 2 + \sqrt{x - 1}$ , you need to swap the roles of $x$ and $y$ and solve for $y$ . Let $y = f(x) = 2 + \sqrt{x - 1}$ . To find the inverse function, you need to solve for $x$ in terms of $y$ .

**Step 1:** Swap the roles of $x$ and $y$
$y = 2 + \sqrt{x - 1}$
Step 2: Subtract 2 from both sides
 $y - 2 = \sqrt{x - 1}$ 
Step 3: Square both sides
 $(y - 2)^2 = x - 1$ 
Step 4: Add 1 to both sides
 $(y - 2)^2 + 1 = x$ 
Step 5: Simplify
 $x = (y - 2)^2 + 1$

The inverse function of $f(x)$ is $f^{-1}(x) = (x - 2)^2 + 1$ .

Q: How do I graph the inverse function $f^{-1}(x) = (x - 2)^2 + 1$ ?

A: To graph the inverse function $f^{-1}(x) = (x - 2)^2 + 1$ , you need to understand its behavior. The inverse function is a reflection of the original function across the line $y = x$ . Since the original function is a curve that starts at $x = 1$ and increases as $x$ increases, the inverse function will be a curve that starts at $y = 1$ and increases as $y$ increases.

**Domain:** $x \geq 1$
**Range:** $y \geq 1$

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

A: The original function and its inverse are related by the fact that they are reflections of each other across the line $y = x$ . This means that if you have a point $(x, y)$ on the graph of the original function, the corresponding point on the graph of the inverse function will be $(y, x)$ .

Q: How do I use the inverse function in real-world applications?

A: The inverse function can be used in a variety of real-world applications, such as:

Modeling population growth: The inverse function can be used to model the growth of a population over time.
Analyzing data: The inverse function can be used to analyze data and identify patterns and trends.
Solving equations: The inverse function can be used to solve equations and find the values of unknown variables.

Conclusion

In this article, we have answered some frequently asked questions about the function $f(x) = 2 + \sqrt{x - 1}$ and its inverse. We have discussed the domain and range of the function, how to find the inverse function, and how to graph the inverse function. We have also discussed the relationship between the original function and its inverse, and how to use the inverse function in real-world applications.