If $g(x)=\frac{x+1}{x-2}$ And $h(x)=4-x$, What Is The Value Of $(g \circ H)(-3)$?A. \$\frac{8}{5}$[/tex\] B. $\frac{5}{2}$ C. $\frac{15}{2}$ D. \$\frac{18}{5}$[/tex\]
Introduction
In mathematics, function composition is a fundamental concept that allows us to combine two or more functions to create a new function. This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis. In this article, we will explore the concept of function composition and how to evaluate the composition of two given functions.
What is Function Composition?
Function composition is the process 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, denoted as (f ∘ g)(x), 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.
Evaluating Function Composition
To evaluate the composition of two functions, we need to follow the order of operations. First, we apply the inner function (g(x)) to the input x, and then apply the outer function (f(x)) to the result.
Example: Evaluating (g ∘ h)(-3)
Let's consider the given functions:
g(x) = (x+1)/(x-2) h(x) = 4-x
We are asked to evaluate the composition (g ∘ h)(-3). To do this, we need to follow the order of operations.
Step 1: Evaluate h(-3)
First, we need to evaluate the inner function h(-3).
h(-3) = 4 - (-3) = 4 + 3 = 7
Step 2: Evaluate g(h(-3))
Now that we have the value of h(-3), we can evaluate the outer function g(h(-3)).
g(h(-3)) = 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
Function composition is a powerful tool in mathematics that allows us to create new functions from existing ones. By understanding how to evaluate function composition, we can solve a wide range of problems in algebra, calculus, and analysis.
Tips and Tricks
- When evaluating function composition, always follow the order of operations.
- Make sure to evaluate the inner function first, and then the outer function.
- Use algebraic manipulations to simplify the expression and make it easier to evaluate.
Conclusion
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about function composition.
Q: What is function composition?
A: Function composition is the process 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, denoted as (f ∘ g)(x), is defined as:
(f ∘ g)(x) = f(g(x))
Q: How do I evaluate function composition?
A: To evaluate function composition, you need to follow the order of operations. First, you apply the inner function (g(x)) to the input x, and then apply the outer function (f(x)) to the result.
Q: What is the difference between function composition and function evaluation?
A: Function evaluation is the process of finding the value of a function at a given input. Function composition, on the other hand, is the process of combining two or more functions to create a new function.
Q: Can I compose more than two functions?
A: Yes, you can compose more than two functions. For example, if you have three functions f(x), g(x), and h(x), you can compose them as (f ∘ g ∘ h)(x) = f(g(h(x))).
Q: How do I know which function to apply first?
A: When evaluating function composition, you need to follow the order of operations. This means that you apply the inner function first, and then the outer function.
Q: Can I use function composition to solve equations?
A: Yes, you can use function composition to solve equations. By applying function composition to both sides of an equation, you can simplify the equation and solve for the unknown variable.
Q: What are some common applications of function composition?
A: Function composition has many applications in mathematics, including:
- Algebra: Function composition is used to solve equations and simplify expressions.
- Calculus: Function composition is used to find derivatives and integrals.
- Analysis: Function composition is used to study the properties of functions and their behavior.
Q: How do I practice function composition?
A: You can practice function composition by working through examples and exercises. You can also try composing different functions to see how they interact.
Q: What are some common mistakes to avoid when evaluating function composition?
A: Some common mistakes to avoid when evaluating function composition include:
- Not following the order of operations
- Not simplifying the expression before evaluating it
- Not checking for domain restrictions
Conclusion
In conclusion, function composition is a powerful tool in mathematics that allows us to create new functions from existing ones. By understanding how to evaluate function composition, we can solve a wide range of problems in algebra, calculus, and analysis. With practice and patience, you can become proficient in evaluating function composition and tackle even the most challenging problems.
Additional Resources
- Khan Academy: Function Composition
- Mathway: Function Composition
- Wolfram Alpha: Function Composition
Practice Problems
- Evaluate (f ∘ g)(x) given f(x) = 2x + 1 and g(x) = x^2 + 1.
- Find the value of (g ∘ f)(-2) given g(x) = x^2 + 1 and f(x) = 2x + 1.
- Simplify the expression (f ∘ g)(x) given f(x) = x^2 + 1 and g(x) = 2x + 1.
Answer Key
- (f ∘ g)(x) = 2(x^2 + 1) + 1 = 2x^2 + 3
- (g ∘ f)(-2) = (2(-2) + 1)^2 + 1 = 3^2 + 1 = 10
- (f ∘ g)(x) = (2x + 1)^2 + 1 = 4x^2 + 4x + 2