If $f(x) = 3x^2$ And $g(x) = 4x^3 + 1$, What Is The Degree Of $(f \circ G)(x$\]?A. 2 B. 3 C. 5 D. 6
In mathematics, the composition of functions is a fundamental concept that allows us to combine 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)). This means that 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:
- f(x) = 3x^2
- g(x) = 4x^3 + 1
We are asked to find the degree of the composition (f ∘ g)(x).
The Composition of Functions
To find the composition (f ∘ g)(x), we need to substitute g(x) into f(x) in place of x. This gives us:
(f ∘ g)(x) = f(g(x)) = 3(g(x))^2 = 3(4x^3 + 1)^2
Expanding the Composition
To find the degree of the composition, we need to expand the expression (4x^3 + 1)^2. Using the binomial theorem, we can expand this expression as follows:
(4x^3 + 1)^2 = (4x3)2 + 2(4x^3)(1) + 1^2 = 16x^6 + 8x^3 + 1
The Degree of the Composition
The degree of a polynomial is the highest power of the variable (in this case, x) that appears in the polynomial. In the expanded expression 16x^6 + 8x^3 + 1, the highest power of x is 6. Therefore, the degree of the composition (f ∘ g)(x) is 6.
Conclusion
In conclusion, the degree of the composition (f ∘ g)(x) is 6. This means that the highest power of x that appears in the composition is 6.
Answer
The correct answer is D. 6.
Additional Examples
To further illustrate the concept of the composition of functions, let's consider a few additional examples.
Example 1
Given f(x) = 2x + 1 and g(x) = x^2 - 3, find the degree of (f ∘ g)(x).
To find the composition (f ∘ g)(x), we need to substitute g(x) into f(x) in place of x. This gives us:
(f ∘ g)(x) = f(g(x)) = 2(g(x)) + 1 = 2(x^2 - 3) + 1 = 2x^2 - 6 + 1 = 2x^2 - 5
The degree of the composition (f ∘ g)(x) is 2.
Example 2
Given f(x) = x^2 + 2 and g(x) = 3x - 1, find the degree of (f ∘ g)(x).
To find the composition (f ∘ g)(x), we need to substitute g(x) into f(x) in place of x. This gives us:
(f ∘ g)(x) = f(g(x)) = (g(x))^2 + 2 = (3x - 1)^2 + 2 = (9x^2 - 6x + 1) + 2 = 9x^2 - 6x + 3
The degree of the composition (f ∘ g)(x) is 2.
Example 3
Given f(x) = x^3 + 1 and g(x) = 2x^2 - 1, find the degree of (f ∘ g)(x).
To find the composition (f ∘ g)(x), we need to substitute g(x) into f(x) in place of x. This gives us:
(f ∘ g)(x) = f(g(x)) = (g(x))^3 + 1 = (2x^2 - 1)^3 + 1 = (8x^6 - 12x^4 + 6x^2 - 1) + 1 = 8x^6 - 12x^4 + 6x^2
The degree of the composition (f ∘ g)(x) is 6.
Conclusion
In conclusion, the degree of the composition (f ∘ g)(x) is 6. This means that the highest power of x that appears in the composition is 6.
Answer
Q: What is the composition of functions?
A: The 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, denoted as (f ∘ g)(x), is defined as f(g(x)). This means that we first apply the function g to the input x, and then apply the function f to the result.
Q: How do I find the composition of two functions?
A: To find the composition of two functions, you need to substitute the second function into the first function in place of x. For example, if we have f(x) = 3x^2 and g(x) = 4x^3 + 1, then the composition (f ∘ g)(x) is found by substituting g(x) into f(x) in place of x.
Q: What is the degree of a composition of functions?
A: The degree of a composition of functions is the highest power of the variable (in this case, x) that appears in the composition. To find the degree of a composition, you need to expand the expression and identify the highest power of x.
Q: How do I expand a composition of functions?
A: To expand a composition of functions, you can use the binomial theorem or simply multiply out the terms. For example, if we have (4x^3 + 1)^2, we can expand it as follows:
(4x^3 + 1)^2 = (4x3)2 + 2(4x^3)(1) + 1^2 = 16x^6 + 8x^3 + 1
Q: What is the difference between the composition of functions and the product of functions?
A: The composition of functions and the product of functions are two different operations. The composition of functions is defined as f(g(x)), while the product of functions is defined as f(x)g(x). For example, if we have f(x) = 3x^2 and g(x) = 4x^3 + 1, then the composition (f ∘ g)(x) is found by substituting g(x) into f(x) in place of x, while the product f(x)g(x) is found by multiplying f(x) and g(x) together.
Q: Can I have multiple compositions of functions?
A: Yes, you can have multiple compositions of functions. For example, if we have f(x) = 3x^2 and g(x) = 4x^3 + 1, then we can find the composition (f ∘ g)(x) and then find the composition (f ∘ (f ∘ g))(x).
Q: How do I find the inverse of a composition of functions?
A: To find the inverse of a composition of functions, you need to swap the functions and then find the inverse of the resulting function. For example, if we have (f ∘ g)(x) = f(g(x)), then the inverse of (f ∘ g)(x) is (g ∘ f)(x) = g(f(x)).
Q: Can I have a composition of functions with multiple variables?
A: Yes, you can have a composition of functions with multiple variables. For example, if we have f(x, y) = 3x^2 + 2y and g(x, y) = 4x^3 + 1, then we can find the composition (f ∘ g)(x, y) by substituting g(x, y) into f(x, y) in place of x and y.
Conclusion
In conclusion, the composition of functions is a powerful tool that allows us to combine two or more functions to create a new function. By understanding how to find the composition of functions, we can solve a wide range of problems in mathematics and other fields.