Consider The Functions F F F And G G G .$ \begin{align*} f(x) &= 4x^2 + 1 \ g(x) &= X^2 - 3 \end{align*} }$Perform The Function Compositions 1. { (f \circ G)(x) = F(g(x))$ 2. \[ 2. \[ 2. \[ (g \circ F)(x) =

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Function Composition: Understanding the Composition of Functions f and g

In mathematics, function composition is a process of combining two or more functions to create a new function. This new function takes the input from the original functions and produces an output based on the rules of the individual functions. In this article, we will explore the composition of two functions, f(x) and g(x), and perform the function compositions (f ∘ g)(x) and (g ∘ f)(x).

Before we proceed with the function compositions, let's define the functions f(x) and g(x).

  • Function f(x): The function f(x) is defined as f(x) = 4x^2 + 1.
  • Function g(x): The function g(x) is defined as g(x) = x^2 - 3.

To perform the function composition (f ∘ g)(x), we need to substitute g(x) into f(x) in place of x.

Step 1: Substitute g(x) into f(x)

We will substitute g(x) = x^2 - 3 into f(x) = 4x^2 + 1.

f(g(x)) = 4(g(x))^2 + 1 f(g(x)) = 4(x^2 - 3)^2 + 1

Step 2: Expand the Equation

Now, let's expand the equation (x^2 - 3)^2.

(x^2 - 3)^2 = x^4 - 6x^2 + 9

Step 3: Substitute the Expanded Equation into f(g(x))

Now, let's substitute the expanded equation into f(g(x)).

f(g(x)) = 4(x^4 - 6x^2 + 9) + 1 f(g(x)) = 4x^4 - 24x^2 + 36 + 1 f(g(x)) = 4x^4 - 24x^2 + 37

Conclusion

The function composition (f ∘ g)(x) is f(g(x)) = 4x^4 - 24x^2 + 37.

To perform the function composition (g ∘ f)(x), we need to substitute f(x) into g(x) in place of x.

Step 1: Substitute f(x) into g(x)

We will substitute f(x) = 4x^2 + 1 into g(x) = x^2 - 3.

g(f(x)) = (f(x))^2 - 3 g(f(x)) = (4x^2 + 1)^2 - 3

Step 2: Expand the Equation

Now, let's expand the equation (4x^2 + 1)^2.

(4x^2 + 1)^2 = 16x^4 + 8x^2 + 1

Step 3: Substitute the Expanded Equation into g(f(x))

Now, let's substitute the expanded equation into g(f(x)).

g(f(x)) = 16x^4 + 8x^2 + 1 - 3 g(f(x)) = 16x^4 + 8x^2 - 2

Conclusion

The function composition (g ∘ f)(x) is g(f(x)) = 16x^4 + 8x^2 - 2.

In this article, we have explored the composition of two functions, f(x) and g(x), and performed the function compositions (f ∘ g)(x) and (g ∘ f)(x). We have seen that the function composition (f ∘ g)(x) is f(g(x)) = 4x^4 - 24x^2 + 37, and the function composition (g ∘ f)(x) is g(f(x)) = 16x^4 + 8x^2 - 2. These results demonstrate the power of function composition in creating new functions from existing ones.