The Functions F F F And G G G Are Defined As F ( X ) = X + 8 F(x) = X + 8 F ( X ) = X + 8 And G ( X ) = ∣ X ∣ G(x) = |x| G ( X ) = ∣ X ∣ .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 F \cdot G F ⋅ G , F ⋅ F F \cdot F F ⋅ F ,

Introduction

In mathematics, functions are used to describe the relationship between variables. The functions $f$ and $g$ are defined as $f(x) = x + 8$ and $g(x) = |x|$. In this article, we will analyze the domain of these functions and their compositions, including $f + g$, $f - g$, $f \cdot g$, and $f \cdot f$.

Domain of f

The function $f(x) = x + 8$ is a linear function, which means it is defined for all real numbers. The domain of $f$ is the set of all real numbers, denoted as $\mathbb{R}$. This is because the expression $x + 8$ is valid for any value of $x$.

Domain of g

The function $g(x) = |x|$ is defined as the absolute value of $x$. The absolute value function is defined for all real numbers, but it has a different behavior for positive and negative values of $x$. For $x \geq 0$, $|x| = x$, and for $x < 0$, $|x| = -x$. Therefore, the domain of $g$ is also the set of all real numbers, $\mathbb{R}$.

Domain of f + g

The function $f + g$ is defined as $(f + g)(x) = f(x) + g(x) = (x + 8) + |x|$. Since the domain of $f$ is $\mathbb{R}$ and the domain of $g$ is $\mathbb{R}$, the domain of $f + g$ is also $\mathbb{R}$. This is because the expression $(x + 8) + |x|$ is valid for any value of $x$.

Domain of f - g

The function $f - g$ is defined as $(f - g)(x) = f(x) - g(x) = (x + 8) - |x|$. Since the domain of $f$ is $\mathbb{R}$ and the domain of $g$ is $\mathbb{R}$, the domain of $f - g$ is also $\mathbb{R}$. This is because the expression $(x + 8) - |x|$ is valid for any value of $x$.

Domain of f * g

The function $f \cdot g$ is defined as $(f \cdot g)(x) = f(x) \cdot g(x) = (x + 8) \cdot |x|$. Since the domain of $f$ is $\mathbb{R}$ and the domain of $g$ is $\mathbb{R}$, the domain of $f \cdot g$ is also $\mathbb{R}$. However, we need to consider the behavior of the absolute value function. When $x < 0$, $|x| = -x$, and when $x \geq 0$, $|x| = x$. Therefore, the expression $(x + 8) \cdot |x|$ is valid for any value of $x$, but we need to consider the case when $x = 0$. When $x = 0$, the expression $(x + 8) \cdot |x|$ is equal to $0 \cdot |0| = 0$. Therefore, the domain of $f \cdot g$ is the set of all real numbers, $\mathbb{R}$.

Domain of f * f

The function $f \cdot f$ is defined as $(f \cdot f)(x) = f(x) \cdot f(x) = (x + 8) \cdot (x + 8)$. Since the domain of $f$ is $\mathbb{R}$, the domain of $f \cdot f$ is also $\mathbb{R}$. This is because the expression $(x + 8) \cdot (x + 8)$ is valid for any value of $x$.

Conclusion

In conclusion, the domain of $f$, $g$, $f + g$, $f - g$, $f \cdot g$, and $f \cdot f$ is the set of all real numbers, $\mathbb{R}$. This is because the functions $f$ and $g$ are defined for all real numbers, and the compositions of these functions are also defined for all real numbers.

References

• [1] "Functions" by Wolfram MathWorld
• [2] "Absolute Value" by Wolfram MathWorld

Further Reading

• [1] "Domain of a Function" by Math Open Reference
• [2] "Composition of Functions" by Math Open Reference
  

Introduction

In our previous article, we analyzed the domain of the functions $f$ and $g$, as well as their compositions, including $f + g$, $f - g$, $f \cdot g$, and $f \cdot f$. In this article, we will answer some frequently asked questions about these functions.

Q: What is the domain of f(x) = x + 8?

A: The domain of $f(x) = x + 8$ is the set of all real numbers, $\mathbb{R}$. This is because the expression $x + 8$ is valid for any value of $x$.

Q: What is the domain of g(x) = |x|?

A: The domain of $g(x) = |x|$ is the set of all real numbers, $\mathbb{R}$. This is because the absolute value function is defined for all real numbers.

Q: What is the domain of f + g?

A: The domain of $f + g$ is the set of all real numbers, $\mathbb{R}$. This is because the expression $(x + 8) + |x|$ is valid for any value of $x$.

Q: What is the domain of f - g?

A: The domain of $f - g$ is the set of all real numbers, $\mathbb{R}$. This is because the expression $(x + 8) - |x|$ is valid for any value of $x$.

Q: What is the domain of f * g?

A: The domain of $f \cdot g$ is the set of all real numbers, $\mathbb{R}$. However, we need to consider the behavior of the absolute value function. When $x < 0$, $|x| = -x$, and when $x \geq 0$, $|x| = x$. Therefore, the expression $(x + 8) \cdot |x|$ is valid for any value of $x$, but we need to consider the case when $x = 0$. When $x = 0$, the expression $(x + 8) \cdot |x|$ is equal to $0 \cdot |0| = 0$. Therefore, the domain of $f \cdot g$ is the set of all real numbers, $\mathbb{R}$.

Q: What is the domain of f * f?

A: The domain of $f \cdot f$ is the set of all real numbers, $\mathbb{R}$. This is because the expression $(x + 8) \cdot (x + 8)$ is valid for any value of $x$.

Q: Can you provide examples of how to find the domain of a function?

A: Yes, here are some examples:

• Find the domain of $f(x) = \frac{1}{x}$. The domain of this function is the set of all real numbers except $0$, because division by zero is undefined.
• Find the domain of $g(x) = \sqrt{x}$. The domain of this function is the set of all non-negative real numbers, because the square root of a negative number is undefined.
• Find the domain of $h(x) = \frac{x^2 - 4}{x - 2}$. The domain of this function is the set of all real numbers except $2$, because the denominator cannot be zero.

Q: Can you provide examples of how to find the domain of a composition of functions?

A: Yes, here are some examples:

• Find the domain of $(f \cdot g)(x) = f(g(x)) = (|x| + 8) \cdot |x|$. The domain of this function is the set of all real numbers, because the expression $(|x| + 8) \cdot |x|$ is valid for any value of $x$.
• Find the domain of $(f + g)(x) = f(x) + g(x) = (x + 8) + |x|$. The domain of this function is the set of all real numbers, because the expression $(x + 8) + |x|$ is valid for any value of $x$.

Conclusion

In conclusion, we have answered some frequently asked questions about the functions $f$ and $g$, as well as their compositions. We have also provided examples of how to find the domain of a function and a composition of functions.

References

• [1] "Functions" by Wolfram MathWorld
• [2] "Absolute Value" by Wolfram MathWorld
• [3] "Domain of a Function" by Math Open Reference
• [4] "Composition of Functions" by Math Open Reference