Let F ( X ) = X − 4 F(x) = X - 4 F ( X ) = X − 4 And G ( X ) = 3 X − 4 G(x) = 3x - 4 G ( X ) = 3 X − 4 .Find F G \frac{f}{g} G F ​ And Use Interval Notation For Its Domain. ( F G ) ( X ) = □ \left(\frac{f}{g}\right)(x) = \square ( G F ​ ) ( X ) = □ Domain: □ \square □

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Let's Dive into the World of Functions: Finding fg\frac{f}{g} and Its Domain

In mathematics, functions play a crucial role in representing relationships between variables. When we have two functions, f(x)f(x) and g(x)g(x), we can perform various operations on them, such as addition, subtraction, multiplication, and division. In this article, we will focus on finding the quotient of two functions, fg\frac{f}{g}, and determining its domain using interval notation.

Defining the Functions

Let's start by defining the two functions:

  • f(x)=x−4f(x) = x - 4
  • g(x)=3x−4g(x) = 3x - 4

These functions are linear, meaning they have a constant rate of change. The function f(x)f(x) represents a line with a slope of 1 and a y-intercept of -4, while the function g(x)g(x) represents a line with a slope of 3 and a y-intercept of -4.

Finding the Quotient

To find the quotient of the two functions, fg\frac{f}{g}, we will divide f(x)f(x) by g(x)g(x). This can be done using the following formula:

(fg)(x)=f(x)g(x)=x−43x−4\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x - 4}{3x - 4}

Simplifying the Quotient

We can simplify the quotient by factoring the numerator and denominator:

(fg)(x)=x−43x−4=(x−4)(3x−4)\left(\frac{f}{g}\right)(x) = \frac{x - 4}{3x - 4} = \frac{(x - 4)}{(3x - 4)}

However, we can further simplify the expression by canceling out the common factor of (x−4)(x - 4):

(fg)(x)=x−43x−4=13⋅x−4x−43\left(\frac{f}{g}\right)(x) = \frac{x - 4}{3x - 4} = \frac{1}{3} \cdot \frac{x - 4}{x - \frac{4}{3}}

Domain of the Quotient

The domain of a function is the set of all possible input values for which the function is defined. In the case of the quotient fg\frac{f}{g}, the domain is restricted by the values of xx that make the denominator g(x)g(x) equal to zero.

To find the domain, we need to solve the equation g(x)=0g(x) = 0:

3x−4=03x - 4 = 0

Solving for xx, we get:

x=43x = \frac{4}{3}

This means that the quotient fg\frac{f}{g} is undefined when x=43x = \frac{4}{3}. Therefore, the domain of the quotient is all real numbers except x=43x = \frac{4}{3}.

Interval Notation for the Domain

In interval notation, the domain of the quotient can be represented as:

(−∞,43)∪(43,∞)\left(-\infty, \frac{4}{3}\right) \cup \left(\frac{4}{3}, \infty\right)

This notation indicates that the quotient is defined for all real numbers except x=43x = \frac{4}{3}.

In conclusion, we have found the quotient of the two functions, fg\frac{f}{g}, and determined its domain using interval notation. The quotient is defined as 13⋅x−4x−43\frac{1}{3} \cdot \frac{x - 4}{x - \frac{4}{3}}, and its domain is all real numbers except x=43x = \frac{4}{3}. This demonstrates the importance of considering the domain of a function when performing operations on it.

The final answer is:

(fg)(x)=13⋅x−4x−43\left(\frac{f}{g}\right)(x) = \frac{1}{3} \cdot \frac{x - 4}{x - \frac{4}{3}}

Domain: (−∞,43)∪(43,∞)\left(-\infty, \frac{4}{3}\right) \cup \left(\frac{4}{3}, \infty\right)
Let's Dive into the World of Functions: Finding fg\frac{f}{g} and Its Domain - Q&A

In our previous article, we explored the concept of finding the quotient of two functions, fg\frac{f}{g}, and determining its domain using interval notation. We defined the functions f(x)=x−4f(x) = x - 4 and g(x)=3x−4g(x) = 3x - 4, and found the quotient to be 13⋅x−4x−43\frac{1}{3} \cdot \frac{x - 4}{x - \frac{4}{3}}. We also determined the domain of the quotient to be all real numbers except x=43x = \frac{4}{3}.

In this article, we will address some common questions and concerns related to finding the quotient of two functions and determining its domain.

Q: What is the difference between a function and a quotient?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A quotient, on the other hand, is the result of dividing one function by another.

Q: How do I know if a quotient is defined or undefined?

A: A quotient is defined if the denominator is not equal to zero. If the denominator is equal to zero, the quotient is undefined.

Q: Can I simplify a quotient by canceling out common factors?

A: Yes, you can simplify a quotient by canceling out common factors. However, be careful not to cancel out any factors that would make the denominator equal to zero.

Q: How do I determine the domain of a quotient?

A: To determine the domain of a quotient, you need to find the values of xx that make the denominator equal to zero. These values are not included in the domain of the quotient.

Q: Can I use interval notation to represent the domain of a quotient?

A: Yes, you can use interval notation to represent the domain of a quotient. For example, if the domain of a quotient is all real numbers except x=43x = \frac{4}{3}, you can represent it as (−∞,43)∪(43,∞)\left(-\infty, \frac{4}{3}\right) \cup \left(\frac{4}{3}, \infty\right).

Q: What if I have a quotient with a variable in the denominator?

A: If you have a quotient with a variable in the denominator, you need to find the values of the variable that make the denominator equal to zero. These values are not included in the domain of the quotient.

Q: Can I use the quotient rule to find the derivative of a quotient?

A: Yes, you can use the quotient rule to find the derivative of a quotient. The quotient rule states that if f(x)f(x) and g(x)g(x) are two functions, then the derivative of the quotient fg\frac{f}{g} is given by:

(fg)′=f′g−fg′g2\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}

In conclusion, finding the quotient of two functions and determining its domain is an important concept in mathematics. By understanding the difference between a function and a quotient, and by knowing how to simplify and determine the domain of a quotient, you can apply these concepts to a wide range of problems.

The final answer is:

  • The quotient of two functions is defined as fg\frac{f}{g}.
  • The domain of a quotient is all real numbers except the values that make the denominator equal to zero.
  • You can use interval notation to represent the domain of a quotient.
  • You can use the quotient rule to find the derivative of a quotient.
  • Not canceling out common factors when simplifying a quotient.
  • Not including the values that make the denominator equal to zero in the domain of a quotient.
  • Not using interval notation to represent the domain of a quotient.
  • Always check if the denominator is equal to zero before simplifying a quotient.
  • Use interval notation to represent the domain of a quotient.
  • Use the quotient rule to find the derivative of a quotient.
  1. Find the quotient of the two functions f(x)=2x−1f(x) = 2x - 1 and g(x)=x+2g(x) = x + 2.
  2. Determine the domain of the quotient fg\frac{f}{g}, where f(x)=x2−4f(x) = x^2 - 4 and g(x)=x−2g(x) = x - 2.
  3. Simplify the quotient fg\frac{f}{g}, where f(x)=x2−4f(x) = x^2 - 4 and g(x)=x−2g(x) = x - 2.
  1. fg=2x−1x+2\frac{f}{g} = \frac{2x - 1}{x + 2}
  2. The domain of the quotient is all real numbers except x=2x = 2.
  3. fg=x+2x−2\frac{f}{g} = \frac{x + 2}{x - 2}