Determine The Domain Of ( G ∘ F ) ( X (g \circ F)(x ( G ∘ F ) ( X ] If F ( X ) = X 2 + 2 X − 4 F(x) = X^2 + 2x - 4 F ( X ) = X 2 + 2 X − 4 And G ( X ) = 1 X + 1 G(x) = \frac{1}{x+1} G ( X ) = X + 1 1 .A. { X ∈ R } \{x \in \mathbb{R}\} { X ∈ R } B. { X ∈ R ∣ X ≠ − 1 } \{x \in \mathbb{R} \mid X \neq -1\} { X ∈ R ∣ X = − 1 } C. ${x \in \mathbb{R} \mid X \neq -3,
Introduction
In mathematics, a composite function is a function that is derived from two or more functions. The process of combining functions is called function composition. When we have two functions, f(x) and g(x), the composite function (g ∘ f)(x) is defined as g(f(x)). In this article, we will determine the domain of the composite function (g ∘ f)(x) given the functions f(x) = x^2 + 2x - 4 and g(x) = 1/(x+1).
Understanding the Domain of a Function
The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function produces a real number as output. For a function to be defined, the denominator of the function must not be equal to zero.
Determining the Domain of f(x)
The function f(x) = x^2 + 2x - 4 is a quadratic function, and its domain is all real numbers. This is because the quadratic function is defined for all real values of x.
Determining the Domain of g(x)
The function g(x) = 1/(x+1) is a rational function, and its domain is all real numbers except x = -1. This is because the denominator of the function cannot be equal to zero.
Determining the Domain of (g ∘ f)(x)
To determine the domain of the composite function (g ∘ f)(x), we need to find the values of x for which the function is defined. Since the function g(x) is not defined at x = -1, we need to find the values of x for which f(x) is not equal to -1.
Solving the Equation f(x) = -1
To find the values of x for which f(x) is not equal to -1, we need to solve the equation f(x) = -1. Substituting f(x) = x^2 + 2x - 4 into the equation, we get:
x^2 + 2x - 4 = -1
Simplifying the equation, we get:
x^2 + 2x - 3 = 0
Factoring the quadratic equation, we get:
(x + 3)(x - 1) = 0
Solving for x, we get:
x + 3 = 0 or x - 1 = 0
x = -3 or x = 1
Therefore, the values of x for which f(x) is not equal to -1 are x = -3 and x = 1.
Determining the Domain of (g ∘ f)(x)
Since the function g(x) is not defined at x = -1, we need to exclude the values of x for which f(x) is equal to -1 from the domain of (g ∘ f)(x). Therefore, the domain of (g ∘ f)(x) is all real numbers except x = -3 and x = 1.
Conclusion
In conclusion, the domain of the composite function (g ∘ f)(x) is all real numbers except x = -3 and x = 1. This is because the function g(x) is not defined at x = -1, and we need to exclude the values of x for which f(x) is equal to -1 from the domain of (g ∘ f)(x).
Answer
Introduction
In our previous article, we determined the domain of the composite function (g ∘ f)(x) given the functions f(x) = x^2 + 2x - 4 and g(x) = 1/(x+1). In this article, we will answer some frequently asked questions related to determining the domain of a composite function.
Q: What is the domain of a composite function?
A: The domain of a composite function is the set of all possible input values for which the function is defined. In other words, it is the set of all possible x-values for which the function produces a real number as output.
Q: How do I determine the domain of a composite function?
A: To determine the domain of a composite function, you need to find the values of x for which the function is defined. This involves finding the values of x for which the denominator of the function is not equal to zero.
Q: What is the difference between the domain of a function and the domain of a composite function?
A: The domain of a function is the set of all possible input values for which the function is defined. The domain of a composite function is the set of all possible input values for which the composite function is defined.
Q: How do I find the values of x for which the denominator of a function is not equal to zero?
A: To find the values of x for which the denominator of a function is not equal to zero, you need to solve the equation denominator = 0. This will give you the values of x for which the denominator is equal to zero, and you can exclude these values from the domain of the function.
Q: What is the significance of the domain of a composite function?
A: The domain of a composite function is significant because it determines the set of all possible input values for which the function is defined. This is important because it helps us to understand the behavior of the function and to make predictions about its output.
Q: Can the domain of a composite function be empty?
A: Yes, the domain of a composite function can be empty. This occurs when the denominator of the function is always equal to zero, or when the function is undefined for all possible input values.
Q: How do I determine the domain of a composite function with multiple functions?
A: To determine the domain of a composite function with multiple functions, you need to find the values of x for which each function is defined. Then, you need to find the intersection of these sets of values to determine the domain of the composite function.
Q: What is the relationship between the domain of a function and the range of a function?
A: The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values of the function. The domain and range of a function are related in that the domain of a function determines the set of all possible input values for which the function produces a real number as output.
Conclusion
In conclusion, determining the domain of a composite function is an important step in understanding the behavior of the function and making predictions about its output. By following the steps outlined in this article, you can determine the domain of a composite function and gain a deeper understanding of the function's behavior.
Answer
The correct answers to the questions are:
- The domain of a composite function is the set of all possible input values for which the function is defined.
- To determine the domain of a composite function, you need to find the values of x for which the function is defined.
- The domain of a composite function is the set of all possible input values for which the composite function is defined.
- To find the values of x for which the denominator of a function is not equal to zero, you need to solve the equation denominator = 0.
- The domain of a composite function is significant because it determines the set of all possible input values for which the function is defined.
- Yes, the domain of a composite function can be empty.
- To determine the domain of a composite function with multiple functions, you need to find the values of x for which each function is defined and then find the intersection of these sets of values.
- The domain of a function is the set of all possible input values for which the function is defined, while the range of a function is the set of all possible output values of the function.