If $g(x)=\frac{x+1}{x-2}$ And $h(x)=4-x$, What Is The Value Of $(g \circ H)(-3)$?A. $ 8 5 \frac{8}{5} 5 8 [/tex] B. $\frac{5}{2}$ C. $\frac{15}{2}$ D. $ 18 5 \frac{18}{5} 5 18 [/tex]
Introduction
In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). Composition of functions is a way of combining two or more functions to create a new function. In this article, we will explore the concept of composition of functions and use it to solve a problem involving two given functions, g(x) and h(x).
What is Composition of Functions?
Composition of functions is a way of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g is denoted by (f ∘ g)(x) and is defined as:
(f ∘ g)(x) = f(g(x))
In other words, we first apply the function g to the input x, and then apply the function f to the result.
The Given Functions
In this problem, we are given two functions:
g(x) = (x + 1) / (x - 2)
h(x) = 4 - x
We are asked to find the value of (g ∘ h)(-3).
Step 1: Apply the Function h to the Input -3
To find the value of (g ∘ h)(-3), we first need to apply the function h to the input -3. This means we need to substitute x = -3 into the function h(x) = 4 - x.
h(-3) = 4 - (-3) = 4 + 3 = 7
Step 2: Apply the Function g to the Result
Now that we have the result of applying the function h to the input -3, we need to apply the function g to this result. This means we need to substitute x = 7 into the function g(x) = (x + 1) / (x - 2).
g(7) = (7 + 1) / (7 - 2) = 8 / 5
Conclusion
Therefore, the value of (g ∘ h)(-3) is 8/5.
Answer
The correct answer is:
A. 8/5
Discussion
Composition of functions is a powerful tool in mathematics that allows us to create new functions by combining existing ones. In this problem, we used the concept of composition of functions to find the value of (g ∘ h)(-3). We first applied the function h to the input -3, and then applied the function g to the result. This allowed us to create a new function that takes the input -3 and produces the output 8/5.
Real-World Applications
Composition of functions has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the composition of functions can be used to model the motion of objects under the influence of gravity. In engineering, the composition of functions can be used to design and optimize complex systems. In economics, the composition of functions can be used to model the behavior of economic systems and make predictions about future trends.
Conclusion
Introduction
In our previous article, we explored the concept of composition of functions and used it to solve a problem involving two given functions, g(x) and h(x). In this article, we will answer some frequently asked questions about composition of functions.
Q: What is the difference between composition of functions and function addition?
A: Composition of functions and function addition are two different mathematical operations. Function addition involves adding two or more functions together, whereas composition of functions involves combining two or more functions to create a new function.
Q: How do I know when to use composition of functions?
A: You should use composition of functions when you need to create a new function by combining two or more existing functions. For example, if you have two functions, f(x) and g(x), and you want to create a new function that takes the input x and produces the output f(g(x)), then you should use composition of functions.
Q: Can I use composition of functions with any type of function?
A: Yes, you can use composition of functions with any type of function, including linear, quadratic, polynomial, and rational functions.
Q: How do I evaluate a composition of functions?
A: To evaluate a composition of functions, you need to follow the order of operations. First, you need to apply the inner function to the input, and then you need to apply the outer function to the result.
Q: Can I use composition of functions to solve real-world problems?
A: Yes, you can use composition of functions to solve real-world problems. For example, in physics, you can use composition of functions to model the motion of objects under the influence of gravity. In engineering, you can use composition of functions to design and optimize complex systems.
Q: What are some common mistakes to avoid when using composition of functions?
A: Some common mistakes to avoid when using composition of functions include:
- Not following the order of operations
- Not evaluating the inner function first
- Not using the correct notation for composition of functions
Q: How do I know if a function is a composition of functions?
A: A function is a composition of functions if it can be written in the form (f ∘ g)(x) = f(g(x)), where f and g are two or more functions.
Q: Can I use composition of functions with functions that have different domains and ranges?
A: Yes, you can use composition of functions with functions that have different domains and ranges. However, you need to make sure that the domain of the outer function is the same as the range of the inner function.
Conclusion
In conclusion, composition of functions is a powerful tool in mathematics that allows us to create new functions by combining existing ones. By understanding the concept of composition of functions and how to use it, you can solve a wide range of mathematical problems and apply it to real-world situations.
Common Composition of Functions Notations
- (f ∘ g)(x) = f(g(x))
- (f ∘ g)(x) = f(g(x))
- (f ∘ g)(x) = f(g(x))
Common Composition of Functions Examples
- (f ∘ g)(x) = f(g(x)) = f(2x) = 2x^2
- (f ∘ g)(x) = f(g(x)) = f(x^2) = x^4
- (f ∘ g)(x) = f(g(x)) = f(x^3) = x^9
Composition of Functions Practice Problems
- Find the value of (f ∘ g)(x) if f(x) = 2x and g(x) = x^2.
- Find the value of (f ∘ g)(x) if f(x) = x^2 and g(x) = 2x.
- Find the value of (f ∘ g)(x) if f(x) = x^3 and g(x) = x^2.
Composition of Functions Resources
- Khan Academy: Composition of Functions
- Mathway: Composition of Functions
- Wolfram Alpha: Composition of Functions