If f is bijective and g: Y → X is well-defined, then (g o f)⁻¹ = _____
Understand the Problem
The question is asking to prove or demonstrate a property of the function g when it is stated that f is bijective and how it relates to the concept of the inverse function f(g)⁻¹.
Answer
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$
Answer for screen readers
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$
Steps to Solve
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Understand the composition of functions Given two functions $f: X \to Y$ and $g: Y \to X$, the composition of these functions is denoted as $(g \circ f)$. This means applying function $f$ first and then applying function $g$.
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Use the properties of inverses When functions are bijective, they have well-defined inverses. We denote the inverse of $f$ as $f^{-1}$ and the inverse of $g$ as $g^{-1}$. For a composition of two functions, the inverse can be expressed as: $$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$$
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Apply the inverse property To find $(g \circ f)^{-1}$, follow the property stated: $$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$$ This means first apply $g^{-1}$ and then apply $f^{-1}$ to any input.
$(g \circ f)^{-1} = f^{-1} \circ g^{-1}$
More Information
This result shows how the inverse of a composition of two functions is the composition of the inverses in reverse order. This property is fundamental in function analysis and is crucial when dealing with bijective functions.
Tips
- Not recognizing that the order of the functions matters in composition. The inverses must also be applied in reverse order.
- Assuming that the composition of two non-bijective functions has valid inverses. Ensure both functions are bijective for the inverse to exist.
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