Which Expression Is Equivalent To $(f \cdot G)(5)$?A. F ( 5 ) × G ( 5 F(5) \times G(5 F ( 5 ) × G ( 5 ]B. F ( 5 ) + G ( 5 F(5) + G(5 F ( 5 ) + G ( 5 ]C. 5 ⋅ F ( 5 5 \cdot F(5 5 ⋅ F ( 5 ]D. 5 ⋅ G ( 5 5 \cdot G(5 5 ⋅ G ( 5 ]

Introduction

In mathematics, function composition is a fundamental concept that involves combining two or more functions to create a new function. When we have two functions, f(x) and g(x), we can compose them to form a new function, denoted as (f ∘ g)(x) or (f ⋅ g)(x). In this article, we will explore the concept of function composition and determine which expression is equivalent to (f ⋅ g)(5).

What is Function Composition?

Function composition is a process of combining two or more functions to create a new function. Given two functions, f(x) and g(x), we can compose them to form a new function, denoted as (f ∘ g)(x) or (f ⋅ g)(x). The composition of functions is defined as:

(f ∘ g)(x) = f(g(x))

This means that we first apply the function g(x) to the input x, and then apply the function f(x) to the result.

Understanding the Given Expression

The given expression is (f ⋅ g)(5). This means that we need to find the value of the function f(x) at x = 5, and then multiply it by the value of the function g(x) at x = 5.

Analyzing the Options

Let's analyze each of the given options to determine which one is equivalent to (f ⋅ g)(5).

Option A: $f(5) \times g(5)$

This option suggests that we need to find the value of the function f(x) at x = 5, and then multiply it by the value of the function g(x) at x = 5. This is exactly what we need to do to evaluate the expression (f ⋅ g)(5).

Option B: $f(5) + g(5)$

This option suggests that we need to find the value of the function f(x) at x = 5, and then add it to the value of the function g(x) at x = 5. This is not equivalent to (f ⋅ g)(5), as we need to multiply the values, not add them.

Option C: $5 \cdot f(5)$

This option suggests that we need to multiply the value of the function f(x) at x = 5 by 5. This is not equivalent to (f ⋅ g)(5), as we need to multiply the values of f(x) and g(x), not just f(x) by a constant.

Option D: $5 \cdot g(5)$

This option suggests that we need to multiply the value of the function g(x) at x = 5 by 5. This is not equivalent to (f ⋅ g)(5), as we need to multiply the values of f(x) and g(x), not just g(x) by a constant.

Conclusion

Based on our analysis, we can conclude that the correct answer is Option A: $f(5) \times g(5)$. This is the only option that is equivalent to (f ⋅ g)(5), as it correctly represents the composition of the functions f(x) and g(x) at x = 5.

Example Use Case

Suppose we have two functions, f(x) = 2x and g(x) = x + 1. We can compose these functions to form a new function, denoted as (f ∘ g)(x) or (f ⋅ g)(x). To evaluate this function at x = 5, we need to find the value of the function f(x) at x = 5, and then multiply it by the value of the function g(x) at x = 5.

First, we need to find the value of the function g(x) at x = 5:

g(5) = 5 + 1 = 6

Next, we need to find the value of the function f(x) at x = 5:

f(5) = 2(5) = 10

Finally, we can multiply the values of f(x) and g(x) to get the value of the composed function at x = 5:

(f ⋅ g)(5) = f(5) × g(5) = 10 × 6 = 60

Therefore, the value of the composed function at x = 5 is 60.

Conclusion

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Q: What is function composition?

A: Function composition is a process of combining two or more functions to create a new function. Given two functions, f(x) and g(x), we can compose them to form a new function, denoted as (f ∘ g)(x) or (f ⋅ g)(x). The composition of functions is defined as:

(f ∘ g)(x) = f(g(x))

Q: How do I evaluate a function composition?

A: To evaluate a function composition, you need to follow these steps:

1. Evaluate the inner function g(x) at the given input x.
2. Use the result as the input to the outer function f(x).
3. Evaluate the outer function f(x) at the result from step 1.

Q: What is the difference between function composition and function addition?

A: Function composition and function addition are two different mathematical operations. Function composition involves combining two or more functions to create a new function, while function addition involves adding two or more functions together.

For example, if we have two functions f(x) = 2x and g(x) = x + 1, we can compose them to form a new function (f ∘ g)(x) = f(g(x)) = 2(x + 1) = 2x + 2. On the other hand, if we add the functions f(x) and g(x) together, we get f(x) + g(x) = 2x + x + 1 = 3x + 1.

Q: Can I compose more than two functions together?

A: Yes, you can compose more than two functions together. For example, if we have three functions f(x), g(x), and h(x), we can compose them to form a new function (f ∘ g ∘ h)(x) = f(g(h(x))). This is known as a chain of function compositions.

Q: How do I determine which expression is equivalent to a function composition?

A: To determine which expression is equivalent to a function composition, you need to follow these steps:

1. Identify the inner function g(x) and the outer function f(x).
2. Evaluate the inner function g(x) at the given input x.
3. Use the result as the input to the outer function f(x).
4. Evaluate the outer function f(x) at the result from step 2.

The resulting expression should be equivalent to the original function composition.

Q: Can I use function composition to solve real-world problems?

A: Yes, function composition can be used to solve real-world problems. For example, in physics, we can use function composition to model the motion of an object under the influence of multiple forces. In economics, we can use function composition to model the behavior of a company's revenue under different market conditions.

Q: What are some common applications of function composition?

A: Some common applications of function composition include:

• Modeling the motion of objects under the influence of multiple forces
• Modeling the behavior of a company's revenue under different market conditions
• Analyzing the behavior of complex systems, such as electrical circuits or mechanical systems
• Solving optimization problems, such as finding the minimum or maximum of a function

Q: How do I practice function composition?

A: To practice function composition, you can try the following exercises:

• Evaluate function compositions with different inner and outer functions.
• Use function composition to solve real-world problems, such as modeling the motion of an object or analyzing the behavior of a company's revenue.
• Practice composing multiple functions together to form a chain of function compositions.
• Use online resources, such as calculators or graphing software, to visualize and explore function compositions.