Select The Correct Answer.Consider These Functions:$\[ \begin{align*} f(x) &= -2x - 5 \\ g(x) &= X - 2 \end{align*} \\]Which Graph Shows The Composite Function \[$(f \cdot G)(x)\$\]?A. B.
Composite Functions: Understanding the Graph of (f ∘ g)(x)
In mathematics, composite functions are a crucial concept in understanding the behavior of functions when they are combined. Given two functions, f(x) and g(x), the composite function (f ∘ g)(x) is defined as f(g(x)). In this article, we will explore the concept of composite functions and determine which graph shows the composite function (f ∘ g)(x), where f(x) = -2x - 5 and g(x) = x - 2.
A composite function is a function that is derived from two or more functions. The process of combining functions is called composition. The composite function (f ∘ g)(x) is defined as f(g(x)), which means that the output of the function g(x) is used as the input for the function f(x).
The Composite Function (f ∘ g)(x)
To find the composite function (f ∘ g)(x), we need to substitute g(x) into f(x). In this case, we have:
f(x) = -2x - 5 g(x) = x - 2
Substituting g(x) into f(x), we get:
(f ∘ g)(x) = f(g(x)) = -2(g(x)) - 5 = -2(x - 2) - 5 = -2x + 4 - 5 = -2x - 1
Graphing the Composite Function
To graph the composite function (f ∘ g)(x), we need to understand the behavior of the individual functions f(x) and g(x). The function f(x) = -2x - 5 is a linear function with a negative slope, while the function g(x) = x - 2 is also a linear function with a positive slope.
When we substitute g(x) into f(x), we get a new function (f ∘ g)(x) = -2x - 1, which is also a linear function with a negative slope. The graph of the composite function (f ∘ g)(x) will be a straight line with a negative slope.
Determining the Correct Graph
Now that we have understood the composite function (f ∘ g)(x), we need to determine which graph shows the correct behavior. The graph should be a straight line with a negative slope.
Graph A
Graph A shows a straight line with a positive slope, which is not consistent with the composite function (f ∘ g)(x). Therefore, Graph A is not the correct answer.
Graph B
Graph B shows a straight line with a negative slope, which is consistent with the composite function (f ∘ g)(x). Therefore, Graph B is the correct answer.
In conclusion, the composite function (f ∘ g)(x) is a linear function with a negative slope. The graph of the composite function (f ∘ g)(x) should be a straight line with a negative slope. Based on the analysis, Graph B is the correct answer.
In our previous article, we explored the concept of composite functions and determined which graph shows the composite function (f ∘ g)(x), where f(x) = -2x - 5 and g(x) = x - 2. In this article, we will provide a Q&A guide to help you better understand composite functions and how to work with them.
Q: What is a composite function?
A: A composite function is a function that is derived from two or more functions. The process of combining functions is called composition. The composite function (f ∘ g)(x) is defined as f(g(x)), which means that the output of the function g(x) is used as the input for the function f(x).
Q: How do I find the composite function (f ∘ g)(x)?
A: To find the composite function (f ∘ g)(x), you need to substitute g(x) into f(x). In other words, you need to replace x in f(x) with g(x).
Q: What is the difference between (f ∘ g)(x) and (g ∘ f)(x)?
A: The composite function (f ∘ g)(x) is different from (g ∘ f)(x). The order of the functions matters. When you substitute g(x) into f(x), you get (f ∘ g)(x). When you substitute f(x) into g(x), you get (g ∘ f)(x).
Q: How do I graph the composite function (f ∘ g)(x)?
A: To graph the composite function (f ∘ g)(x), you need to understand the behavior of the individual functions f(x) and g(x). The graph of the composite function (f ∘ g)(x) will be a straight line with a negative slope, if the original functions are linear.
Q: What are some common mistakes to avoid when working with composite functions?
A: Some common mistakes to avoid when working with composite functions include:
- Not substituting g(x) into f(x) correctly
- Not understanding the order of the functions
- Not graphing the composite function correctly
Q: How do I determine which graph shows the composite function (f ∘ g)(x)?
A: To determine which graph shows the composite function (f ∘ g)(x), you need to look for a straight line with a negative slope. If the graph shows a straight line with a positive slope, it is not the correct answer.
Q: Can I use composite functions to solve real-world problems?
A: Yes, composite functions can be used to solve real-world problems. For example, you can use composite functions to model population growth, financial transactions, and other complex systems.
In conclusion, composite functions are a powerful tool for solving complex problems. By understanding how to work with composite functions, you can model real-world systems and make predictions about future outcomes. We hope this Q&A guide has helped you better understand composite functions and how to work with them.
The final answer is that composite functions are a powerful tool for solving complex problems, and by understanding how to work with them, you can model real-world systems and make predictions about future outcomes.