Problem 74
Question
How can a graphing utility be used to visually determine if two functions are inverses of each other?
Step-by-Step Solution
Verified Answer
A graphing utility can help determine if two functions are inverses of each other by checking if the graphs of the two functions are mirror images of each other across the line y=x.
1Step 1: Understanding Inverse Functions
An inverse function is a function that undoes the action of the other function. A function \( f \) and its inverse \(f^{-1}\) are said to be inverses if and only if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) where \( f(f^{-1}(x)) \) is the function complement of the inverse function and \( f^{-1}(f(x)) \) is the inverse function complement of the function.
2Step 2: Reflection concept
The graphs of a function and its inverse are reflections of each other across the line \( y=x \). If a given point, say, (a, b) lies on the plot of the function, then the point (b, a) will lie on the plot of the inverse function.
3Step 3: Plotting the functions
Now, to visually determine if two functions are inverses of each other, first, plot both functions on the same set of axes. Then, also plot the line \(y=x\).
4Step 4: Visually checking the reflection
Inspect the resulting plot. If the graphs of the functions are symmetrical about the line \(y=x\), then they are inverses of each other.
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