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The concept of composition of functions is a fundamental tool in mathematics that allows us to build new functions by combining existing ones. Think of it as a process where one function is applied to the result of another function. For example, if you have two functions, \( f(x) \) and \( g(x) \), the composition \( f \circ g \) is formed by applying \( g \) first and then \( f \) to the result of \( g \). This can be thought of in the following way:

• You start by taking an input \( x \).
• Apply \( g \) to \( x \) to get \( g(x) \).
• Then, take this result and apply \( f \) to it, yielding \( f(g(x)) \).

This step-by-step application is simply referred to as composing the functions. In our exercise, if we first apply \( g(x) = 4x+5 \), and then \( f(x) = x^3 \), we evaluate it as \( f(g(x)) = (4x+5)^3 \). The power behind function composition is its ability to simplify complex problems by breaking them into smaller, manageable parts.

A function is described as one-to-one, or injective, if it maps distinct inputs to distinct outputs. This means, for every different input, the function produces a different output. Understanding one-to-one functions is crucial, especially when dealing with inverse functions.To determine if a function is one-to-one, you can apply the horizontal line test: if no horizontal line intersects the graph of the function more than once, then the function is one-to-one. In the context of the original exercise, both \( f(x) = x^3 \) and \( g(x) = 4x + 5 \) are one-to-one functions:

• For \( f(x) = x^3 \), each value produces a unique cubed result, ensuring distinct outputs for distinct inputs.
• Similarly, \( g(x) = 4x + 5 \) increases linearly and never repeats any output for different inputs.

When a function is one-to-one, an inverse function can be uniquely defined, making it invertible. This property is essential to verify the exercise's proposition involving function inverses.

Function inverses are a way to reverse the action of a function. If you have a function \( f \), and there exists another function \( f^{-1} \) such that applying \( f \) followed by \( f^{-1} \) (or vice versa) leaves you where you started, then \( f^{-1} \) is called the inverse of \( f \).Mathematically, if \( y = f(x) \), then applying the inverse function will give you back your original \( x \) value:

• \( f^{-1}(y) = x \)

In the exercise, the inverse of \( g(x) = 4x + 5 \) is found to be \( g^{-1}(x) = \frac{x - 5}{4} \) and the inverse of \( f(x) = x^3 \) is \( f^{-1}(x) = \sqrt[3]{x} \). These inverses reverse their respective functions, bringing you back to the initial value before the function was applied.The exercise involves verifying a neat property: \((f \circ g)^{-1} = g^{-1} \circ f^{-1}\). Applying both inverses in reverse order brings the equation back to its initial form. This helps in understanding why certain problems can be decomposed into simpler ones and solved sequentially.