Given \[$ F(x) = X + 4 \$\] And \[$ G(x) = \sqrt{x+5} \$\], Determine The Following. Write Each Answer Using Interval Notation.1. Determine The Domain Of \[$ F(g(x)) \$\]. Domain: \[$\square\$\] \[$ (-\infty,
Introduction
In this article, we will explore the concept of composition of functions and determine the domain of the composite function f(g(x)). We will use the given functions f(x) = x + 4 and g(x) = β(x+5) to find the domain of f(g(x)).
Understanding Composition of Functions
Composition of functions is a process of combining two or more functions to create a new function. The output of one function is used as the input for the next function. In this case, we have two functions f(x) and g(x), and we want to find the composite function f(g(x)).
Step 1: Finding the Composite Function f(g(x))
To find the composite function f(g(x)), we need to substitute g(x) into f(x) in place of x. This means we will replace x in f(x) = x + 4 with g(x) = β(x+5).
f(g(x)) = f(β(x+5)) = β(x+5) + 4
Step 2: Determining the Domain of f(g(x))
To determine the domain of f(g(x)), we need to consider the restrictions on the input values of x. The function g(x) = β(x+5) has a restriction that x+5 β₯ 0, because the square root of a negative number is not defined in real numbers.
x+5 β₯ 0 x β₯ -5
This means that the input values of x must be greater than or equal to -5. Now, we need to consider the output values of g(x), which are used as input values for f(x).
The function f(x) = x + 4 has no restrictions on its input values, but it does have a restriction on its output values. The output values of f(x) must be real numbers.
Since the output values of g(x) are used as input values for f(x), we need to ensure that the output values of g(x) are real numbers. The function g(x) = β(x+5) has real output values when x β₯ -5.
However, we also need to consider the restriction on the input values of x, which is x β₯ -5. This means that the domain of f(g(x)) is the set of all real numbers x such that x β₯ -5.
Conclusion
In conclusion, the domain of f(g(x)) is the set of all real numbers x such that x β₯ -5. This can be written in interval notation as:
[-5, β)
Final Answer
Introduction
In our previous article, we explored the concept of composition of functions and determined the domain of the composite function f(g(x)). In this article, we will answer some frequently asked questions about composition of functions and provide additional examples to help you understand the concept better.
Q: What is the difference between composition of functions and function notation?
A: Composition of functions is a process of combining two or more functions to create a new function. Function notation, on the other hand, is a way of writing a function as a mathematical expression. For example, f(x) = x + 4 is a function notation, while f(g(x)) = β(x+5) + 4 is a composition of functions.
Q: How do I determine the domain of a composite function?
A: To determine the domain of a composite function, you need to consider the restrictions on the input values of x. You also need to consider the output values of the inner function, which are used as input values for the outer function. In the case of f(g(x)), we need to ensure that the output values of g(x) are real numbers and that the input values of x are greater than or equal to -5.
Q: Can I have multiple composite functions?
A: Yes, you can have multiple composite functions. For example, if we have two functions f(x) = x + 4 and g(x) = β(x+5), we can create a composite function f(g(x)) = β(x+5) + 4. We can also create another composite function g(f(x)) = β(x+4) + 5.
Q: How do I evaluate a composite function?
A: To evaluate a composite function, you need to follow the order of operations. First, evaluate the inner function, and then use the output values as input values for the outer function. For example, to evaluate f(g(x)) = β(x+5) + 4, we need to first evaluate g(x) = β(x+5), and then use the output values as input values for f(x) = x + 4.
Q: Can I have a composite function with multiple inputs?
A: Yes, you can have a composite function with multiple inputs. For example, if we have two functions f(x,y) = x + y and g(x,y) = β(x+y), we can create a composite function f(g(x,y)) = β(x+y) + x + y.
Q: How do I determine the range of a composite function?
A: To determine the range of a composite function, you need to consider the output values of the inner function, which are used as input values for the outer function. In the case of f(g(x)), we need to ensure that the output values of g(x) are real numbers and that the input values of x are greater than or equal to -5.
Q: Can I have a composite function with a variable as the input?
A: Yes, you can have a composite function with a variable as the input. For example, if we have two functions f(x) = x + 4 and g(x) = β(x+5), we can create a composite function f(g(x)) = β(x+5) + 4, where x is a variable.
Conclusion
In conclusion, composition of functions is a powerful tool for creating new functions from existing ones. By understanding the concept of composition of functions, you can create complex functions and solve problems in mathematics, science, and engineering.
Additional Examples
- Find the composite function f(g(x)) = β(x+5) + 4, where f(x) = x + 4 and g(x) = β(x+5).
- Find the domain of the composite function f(g(x)) = β(x+5) + 4, where f(x) = x + 4 and g(x) = β(x+5).
- Find the range of the composite function f(g(x)) = β(x+5) + 4, where f(x) = x + 4 and g(x) = β(x+5).
- Find the composite function f(g(x)) = β(x+5) + 4, where f(x) = x + 4 and g(x) = β(x+5), and evaluate it at x = 2.
Final Answer
The final answer is that composition of functions is a powerful tool for creating new functions from existing ones. By understanding the concept of composition of functions, you can create complex functions and solve problems in mathematics, science, and engineering.