The Functions F F F And G G G Are Defined As F ( X ) = X − 15 F(x) = \sqrt{x-15} F ( X ) = X − 15 ​ And G ( X ) = X + 15 G(x) = \sqrt{x+15} G ( X ) = X + 15 ​ .a) Find The Domain Of F F F , G G G , F + G F+g F + G , F − G F-g F − G , F G Fg F G , F G \frac{f}{g} G F ​ , And

Introduction

In this article, we will delve into the domain analysis of two given functions, $f(x) = \sqrt{x-15}$ and $g(x) = \sqrt{x+15}$. The domain of a function is the set of all possible input values for which the function is defined. Understanding the domain of a function is crucial in mathematics, as it helps us determine the validity of a function's output. In this discussion, we will explore the domain of $f$, $g$, $f+g$, $f-g$, $fg$, and $\frac{f}{g}$.

Domain of $f(x) = \sqrt{x-15}$

The function $f(x) = \sqrt{x-15}$ is defined when the expression inside the square root is non-negative. This means that $x-15 \geq 0$, which implies that $x \geq 15$. Therefore, the domain of $f(x)$ is $[15, \infty)$.

Domain of $g(x) = \sqrt{x+15}$

The function $g(x) = \sqrt{x+15}$ is defined when the expression inside the square root is non-negative. This means that $x+15 \geq 0$, which implies that $x \geq -15$. Therefore, the domain of $g(x)$ is $[-15, \infty)$.

Domain of $f+g$

The function $f+g$ is defined as $(f+g)(x) = f(x) + g(x) = \sqrt{x-15} + \sqrt{x+15}$. Since both $f(x)$ and $g(x)$ are defined when $x \geq 15$, the domain of $f+g$ is also $[15, \infty)$.

Domain of $f-g$

The function $f-g$ is defined as $(f-g)(x) = f(x) - g(x) = \sqrt{x-15} - \sqrt{x+15}$. Since both $f(x)$ and $g(x)$ are defined when $x \geq 15$, the domain of $f-g$ is also $[15, \infty)$.

Domain of $fg$

The function $fg$ is defined as $(fg)(x) = f(x)g(x) = \sqrt{x-15}\sqrt{x+15} = \sqrt{(x-15)(x+15)}$. Since the expression inside the square root is non-negative when $x \geq -15$, the domain of $fg$ is $[-15, \infty)$.

Domain of $\frac{f}{g}$

The function $\frac{f}{g}$ is defined as $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{\sqrt{x-15}}{\sqrt{x+15}}$. Since both $f(x)$ and $g(x)$ are defined when $x \geq 15$, the domain of $\frac{f}{g}$ is also $[15, \infty)$.

Conclusion

In conclusion, the domain of $f(x) = \sqrt{x-15}$ is $[15, \infty)$, while the domain of $g(x) = \sqrt{x+15}$ is $[-15, \infty)$. The domain of $f+g$, $f-g$, and $\frac{f}{g}$ is $[15, \infty)$, while the domain of $fg$ is $[-15, \infty)$. Understanding the domain of these functions is essential in mathematics, as it helps us determine the validity of a function's output.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Real Analysis, 2nd edition, Richard Royden

Future Work

Q: What is the domain of $f(x) = \sqrt{x-15}$?

A: The domain of $f(x) = \sqrt{x-15}$ is $[15, \infty)$, since the expression inside the square root must be non-negative.

Q: What is the domain of $g(x) = \sqrt{x+15}$?

A: The domain of $g(x) = \sqrt{x+15}$ is $[-15, \infty)$, since the expression inside the square root must be non-negative.

Q: What is the domain of $f+g$?

A: The domain of $f+g$ is $[15, \infty)$, since both $f(x)$ and $g(x)$ are defined when $x \geq 15$.

Q: What is the domain of $f-g$?

A: The domain of $f-g$ is $[15, \infty)$, since both $f(x)$ and $g(x)$ are defined when $x \geq 15$.

Q: What is the domain of $fg$?

A: The domain of $fg$ is $[-15, \infty)$, since the expression inside the square root must be non-negative.

Q: What is the domain of $\frac{f}{g}$?

A: The domain of $\frac{f}{g}$ is $[15, \infty)$, since both $f(x)$ and $g(x)$ are defined when $x \geq 15$.

Q: Can you provide an example of a value in the domain of $f(x) = \sqrt{x-15}$?

A: Yes, an example of a value in the domain of $f(x) = \sqrt{x-15}$ is $x = 20$, since $20 - 15 = 5$ is non-negative.

Q: Can you provide an example of a value in the domain of $g(x) = \sqrt{x+15}$?

A: Yes, an example of a value in the domain of $g(x) = \sqrt{x+15}$ is $x = -10$, since $-10 + 15 = 5$ is non-negative.

Q: What happens if we try to evaluate $f(x)$ or $g(x)$ at a value outside of their domain?

A: If we try to evaluate $f(x)$ or $g(x)$ at a value outside of their domain, we will get an undefined or imaginary result. For example, if we try to evaluate $f(x)$ at $x = 10$, we will get $f(10) = \sqrt{10-15} = \sqrt{-5}$, which is undefined.

Q: Can you provide a graph of the functions $f(x)$ and $g(x)$?

A: Yes, here is a graph of the functions $f(x)$ and $g(x)$:

Note: The graph is not provided here, but it can be easily created using a graphing calculator or software.

Q: What are some real-world applications of the functions $f(x)$ and $g(x)$?

A: The functions $f(x)$ and $g(x)$ have many real-world applications, such as:

• Modeling population growth and decline
• Analyzing financial data
• Studying the behavior of physical systems
• Optimizing complex systems

These are just a few examples, and there are many more applications of the functions $f(x)$ and $g(x)$ in various fields.