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In mathematics, composite functions are a crucial concept in algebra and calculus. A composite function is a function that is derived from the composition of two or more functions. In this article, we will explore how to express composite functions in standard form, using the given examples of functions f(x) and g(x).

Understanding Composite Functions

A composite function is a function that is derived from the composition of two or more functions. For example, if we have two functions f(x) and g(x), the composite function (f ∘ g)(x) is defined as 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.

Expressing f(g(x)) in Standard Form

Let's consider the given example of functions f(x) and g(x):

f(x) = 3x + 14 g(x) = 2x^2 + 3x + 15

To express the composite function (f ∘ g)(x) in standard form, we need to substitute g(x) into f(x) in place of x. This means that we will replace x in the expression for f(x) with the expression for g(x).

(f ∘ g)(x) = f(g(x)) = 3(g(x)) + 14 = 3(2x^2 + 3x + 15) + 14

Now, let's simplify the expression by distributing the 3 to each term inside the parentheses:

(f ∘ g)(x) = 3(2x^2) + 3(3x) + 3(15) + 14 = 6x^2 + 9x + 45 + 14 = 6x^2 + 9x + 59

Therefore, the composite function (f ∘ g)(x) in standard form is:

(f ∘ g)(x) = 6x^2 + 9x + 59

Expressing g(f(x)) in Standard Form

Now, let's consider the composite function (g ∘ f)(x). To express this function in standard form, we need to substitute f(x) into g(x) in place of x.

(g ∘ f)(x) = g(f(x)) = 2(f(x))^2 + 3(f(x)) + 15 = 2(3x + 14)^2 + 3(3x + 14) + 15

Now, let's simplify the expression by expanding the squared term:

(g ∘ f)(x) = 2(9x^2 + 56x + 196) + 3(3x + 14) + 15 = 18x^2 + 112x + 392 + 9x + 42 + 15 = 18x^2 + 121x + 449

Therefore, the composite function (g ∘ f)(x) in standard form is:

(g ∘ f)(x) = 18x^2 + 121x + 449

Conclusion

In this article, we have explored how to express composite functions in standard form. We have used the given examples of functions f(x) and g(x) to demonstrate how to substitute one function into another and simplify the resulting expression. By following these steps, we can express composite functions in standard form and gain a deeper understanding of the relationships between different functions.

Key Takeaways

  • A composite function is a function that is derived from the composition of two or more functions.
  • To express a composite function in standard form, we need to substitute one function into another and simplify the resulting expression.
  • The composite function (f ∘ g)(x) is defined as f(g(x)), where we first apply the function g(x) to the input x and then apply the function f(x) to the result.
  • The composite function (g ∘ f)(x) is defined as g(f(x)), where we first apply the function f(x) to the input x and then apply the function g(x) to the result.

Practice Problems

  1. Find the composite function (f ∘ g)(x) if f(x) = 2x^2 + 3x + 1 and g(x) = x^2 + 2x + 1.
  2. Find the composite function (g ∘ f)(x) if f(x) = x^2 + 2x + 1 and g(x) = 2x^2 + 3x + 1.
  3. Find the composite function (f ∘ g)(x) if f(x) = 3x + 2 and g(x) = 2x^2 + 3x + 1.

Solutions

  1. (f ∘ g)(x) = f(g(x)) = 2(g(x))^2 + 3(g(x)) + 1 = 2(x^2 + 2x + 1)^2 + 3(x^2 + 2x + 1) + 1 = 2(x^4 + 4x^3 + 4x^2 + 1) + 3x^2 + 6x + 3 + 1 = 2x^4 + 8x^3 + 8x^2 + 2 + 3x^2 + 6x + 4 = 2x^4 + 8x^3 + 11x^2 + 6x + 6

  2. (g ∘ f)(x) = g(f(x)) = 2(f(x))^2 + 3(f(x)) + 1 = 2(x^2 + 2x + 1)^2 + 3(x^2 + 2x + 1) + 1 = 2(x^4 + 4x^3 + 4x^2 + 1) + 3x^2 + 6x + 3 + 1 = 2x^4 + 8x^3 + 8x^2 + 2 + 3x^2 + 6x + 4 = 2x^4 + 8x^3 + 11x^2 + 6x + 6

  3. (f ∘ g)(x) = f(g(x)) = 3(g(x)) + 2 = 3(2x^2 + 3x + 1) + 2 = 6x^2 + 9x + 3 + 2 = 6x^2 + 9x + 5
    Composite Functions Q&A ==========================

In this article, we will answer some frequently asked questions about composite functions. Composite functions are a crucial concept in algebra and calculus, and understanding them is essential for solving problems in mathematics.

Q: What is a composite function?

A: A composite function is a function that is derived from the composition of two or more functions. For example, if we have two functions f(x) and g(x), the composite function (f ∘ g)(x) is defined as 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.

Q: How do I express a composite function in standard form?

A: To express a composite function in standard form, we need to substitute one function into another and simplify the resulting expression. For example, if we have the composite function (f ∘ g)(x), we need to substitute g(x) into f(x) in place of x.

Q: What is the difference between (f ∘ g)(x) and (g ∘ f)(x)?

A: The composite function (f ∘ g)(x) is defined as f(g(x)), where we first apply the function g(x) to the input x and then apply the function f(x) to the result. On the other hand, the composite function (g ∘ f)(x) is defined as g(f(x)), where we first apply the function f(x) to the input x and then apply the function g(x) to the result.

Q: How do I determine the order of operations when working with composite functions?

A: When working with composite functions, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I simplify a composite function by canceling out common factors?

A: Yes, you can simplify a composite function by canceling out common factors. For example, if we have the composite function (f ∘ g)(x) = (2x + 1)(x + 2), we can simplify it by canceling out the common factor (x + 1):

(f ∘ g)(x) = (2x + 1)(x + 2) = (2x + 1)(x + 1 + 1) = (2x + 1)(x + 1) + (2x + 1) = 2x^2 + 3x + 2x + 1 = 2x^2 + 5x + 1

Q: How do I determine the domain of a composite function?

A: To determine the domain of a composite function, we need to consider the domains of the individual functions. For example, if we have the composite function (f ∘ g)(x), we need to consider the domain of g(x) and the domain of f(x).

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, if we have a problem that involves a series of operations, we can use composite functions to represent the problem and solve it.

Practice Problems

  1. Find the composite function (f ∘ g)(x) if f(x) = 2x^2 + 3x + 1 and g(x) = x^2 + 2x + 1.
  2. Find the composite function (g ∘ f)(x) if f(x) = x^2 + 2x + 1 and g(x) = 2x^2 + 3x + 1.
  3. Find the domain of the composite function (f ∘ g)(x) if f(x) = 2x^2 + 3x + 1 and g(x) = x^2 + 2x + 1.

Solutions

  1. (f ∘ g)(x) = f(g(x)) = 2(g(x))^2 + 3(g(x)) + 1 = 2(x^2 + 2x + 1)^2 + 3(x^2 + 2x + 1) + 1 = 2(x^4 + 4x^3 + 4x^2 + 1) + 3x^2 + 6x + 3 + 1 = 2x^4 + 8x^3 + 8x^2 + 2 + 3x^2 + 6x + 4 = 2x^4 + 8x^3 + 11x^2 + 6x + 6

  2. (g ∘ f)(x) = g(f(x)) = 2(f(x))^2 + 3(f(x)) + 1 = 2(x^2 + 2x + 1)^2 + 3(x^2 + 2x + 1) + 1 = 2(x^4 + 4x^3 + 4x^2 + 1) + 3x^2 + 6x + 3 + 1 = 2x^4 + 8x^3 + 8x^2 + 2 + 3x^2 + 6x + 4 = 2x^4 + 8x^3 + 11x^2 + 6x + 6

  3. The domain of the composite function (f ∘ g)(x) is the intersection of the domains of f(x) and g(x). Since f(x) = 2x^2 + 3x + 1 and g(x) = x^2 + 2x + 1, the domain of (f ∘ g)(x) is the set of all real numbers x such that x^2 + 2x + 1 > 0 and 2x^2 + 3x + 1 > 0.