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

Verified step by step guidance

1. Identify the function g(x) which is given as g(x) = x^3.
2. Substitute 1/z into the function g(x) in place of x, resulting in g(1/z).
3. Calculate g(1/z) by raising (1/z) to the power of 3, which is (1/z)^3.
4. Simplify the expression (1/z)^3 to 1/z^3.
5. The simplified form of g(1/z) is 1/z^3.

Key Concepts

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In this case, if we have functions f(x) and g(x), the composite function g(f(x)) means we first compute f(x) and then apply g to that result. Understanding how to manipulate and evaluate composite functions is essential for simplifying expressions like g(1/z).

Function Notation

Function notation is a way to denote functions and their inputs clearly. For example, g(x) indicates the function g evaluated at x. In the expression g(1/z), we are substituting 1/z into the function g, which requires understanding how to interpret and apply the notation correctly to find the output.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form, making it easier to work with or evaluate. This can include combining like terms, factoring, or substituting values. In the context of g(1/z), simplifying the expression means substituting 1/z into the function g and then performing any necessary algebraic operations to express the result in a more manageable form.