Composite functions and notation
Let ƒ(x)= x² - 4 , g(x) = x³ and F(x) = 1/(x-3).
Simplify or evaluate the following expressions.
F(g(y))

Verified step by step guidance

1. Identify the functions involved: \( F(x) = \frac{1}{x-3} \) and \( g(x) = x^3 \).
2. Understand the composition of functions: \( F(g(y)) \) means substituting \( g(y) \) into \( F(x) \).
3. Calculate \( g(y) \): Since \( g(x) = x^3 \), then \( g(y) = y^3 \).
4. Substitute \( g(y) = y^3 \) into \( F(x) \): Replace \( x \) in \( F(x) = \frac{1}{x-3} \) with \( y^3 \).
5. Simplify the expression: \( F(g(y)) = \frac{1}{y^3 - 3} \).

Key Concepts

Composite Functions

A composite function is formed when one function is applied to the result of another function. In this case, F(g(y)) means we first evaluate g(y) and then use that result as the input for F. Understanding how to combine functions is crucial for simplifying expressions involving multiple functions.

Function Notation

Function notation is a way to denote functions and their inputs clearly. For example, f(x) indicates that f is a function of x. This notation helps in identifying which function to apply and in what order, especially when dealing with composite functions like F(g(y)).

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to understand or compute. In the context of composite functions, this may include substituting values and performing algebraic operations to combine the functions into a single expression, which is essential for evaluating F(g(y)).