Problem 70
Question
Explain how to determine if two functions are inverses of each other.
Step-by-Step Solution
Verified Answer
Two functions f(x) and g(x) can be confirmed as inverses of each other by verifying if f(g(x)) = x and g(f(x)) = x for all x in their domain.
1Step 1: Understand the Concept of Inverse Functions
An inverse function is a second function which undoes the work of the first one. In other words, if you have a function f(x) and its inverse g(x), applying f and then g will bring you back to your start. Mathematically, this is expressed as f(g(x)) = x and g(f(x)) = x. This property needs to be satisfied for all x in the domain of both functions.
2Step 2: Apply the function f to g(x)
Compute f(g(x)) by substituting g(x) into function f wherever x appears. Simplify the expression and check whether the result is x.
3Step 3: Apply the function g to f(x)
Similarly, compute g(f(x)) by substituting f(x) into function g wherever x appears. Simplify the expression and check whether the result is x.
4Step 4: Verify Conditions
If both f(g(x)) and g(f(x)) simplify to x for every x in the domain of f and g, then the functions are inverses of each other. If these conditions are not met, then f and g are not inverse functions.
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