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


(g o ƒ ) (x) = x⁴ + 3

Key Concepts

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In this case, we are looking for a function ƒ such that when g is applied to ƒ, the result is a new function. This is denoted as (g o ƒ)(x), which means g(ƒ(x)). Understanding how to manipulate and combine functions is crucial for solving the problem.

Quadratic and Polynomial Functions

The function g(x) = x² + 3 is a quadratic function, which is a specific type of polynomial function characterized by its degree of 2. The composition (g o ƒ)(x) results in a polynomial of degree 4, indicating that the function ƒ must also be a polynomial that, when composed with g, yields a higher degree polynomial. Recognizing the properties of polynomial functions helps in determining the form of ƒ.

Finding Inverse Relationships

To find the function ƒ that satisfies the composition (g o ƒ)(x) = x⁴ + 3, we can think of it as finding an inverse relationship. This involves determining what input to g will produce the desired output. By setting g(ƒ(x)) equal to x⁴ + 3, we can derive ƒ(x) by manipulating the equation, which is essential for solving the problem effectively.