Missing piece Let g(x) = x² + 3 Find a function ƒ  that produces the given composition.


(ƒ o g) (x) = x⁴ + 6x² + 9

Verified step by step guidance

1. Step 1: Understand the composition of functions. The composition (f o g)(x) means f(g(x)). We need to find a function f such that when g(x) is plugged into f, it results in the expression x^4 + 6x^2 + 9.
2. Step 2: Identify g(x). We are given that g(x) = x^2 + 3. This is the inner function in the composition.
3. Step 3: Analyze the expression x^4 + 6x^2 + 9. Notice that this expression can be rewritten as (x^2 + 3)^2. This suggests that f(u) = u^2, where u = g(x).
4. Step 4: Verify the composition. Substitute g(x) = x^2 + 3 into f(u) = u^2 to check if it results in the given expression. f(g(x)) = (x^2 + 3)^2 = x^4 + 6x^2 + 9, which matches the given composition.
5. Step 5: Conclude that the function f(x) is x^2. Therefore, f(x) = x^2 is the function that, when composed with g(x), gives the desired result.

Key Concepts

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In this case, if we have functions f and g, the composition (f o g)(x) means we first apply g to x and then apply f to the result of g. Understanding how to manipulate and derive compositions is essential for solving the problem.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c. In this problem, g(x) = x² + 3 is a quadratic function. Recognizing the properties of quadratic functions, such as their parabolic shape and how they can be transformed, is crucial for determining the appropriate function f that will yield the desired composition.

Polynomial Degree

The degree of a polynomial is the highest power of the variable in the expression. In the given composition (ƒ o g)(x) = x⁴ + 6x² + 9, the highest degree is 4, indicating that the function f must be designed to produce a polynomial of this degree when composed with g. Understanding how degrees of polynomials interact during composition helps in constructing the correct function f.