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


(g o ƒ ) (x) = x²⸍³ + 3

Verified step by step guidance

1. Step 1: Understand the composition (g \(\circ\) f)(x) = g(f(x)). We need to find a function f(x) such that when g is applied to f(x), it results in x^{2/3} + 3.
2. Step 2: Recall that g(x) = x^2 + 3. We want g(f(x)) = f(x)^2 + 3 to equal x^{2/3} + 3.
3. Step 3: Set up the equation f(x)^2 + 3 = x^{2/3} + 3.
4. Step 4: Subtract 3 from both sides to isolate the squared term: f(x)^2 = x^{2/3}.
5. Step 5: Solve for f(x) by taking the square root of both sides: f(x) = \(\sqrt{x^{2/3}\)}.

Key Concepts

Function Composition

Function composition involves combining two functions where the output of one function becomes the input of another. In this case, the composition (g o ƒ)(x) means applying function ƒ first and then applying function g to the result. Understanding how to manipulate and combine functions is essential for solving problems involving compositions.

Identifying Functions

To find the function ƒ that satisfies the composition (g o ƒ)(x) = x² + 3, we need to identify the structure of g(x) and how it relates to ƒ. Here, g(x) = x² + 3 suggests that ƒ must produce an input that, when squared and increased by 3, results in the desired output. Recognizing the form of g helps in determining the appropriate form of ƒ.

Inverse Functions

Inverse functions are crucial in understanding how to 'reverse' the operations of a function. If we can express g(x) in terms of its inverse, we can derive ƒ by manipulating the equation. For instance, if we can isolate x in terms of g, we can find the function that, when composed with g, yields the original input, aiding in solving the composition problem.