Work out the inverse function for each equation.
y = x − 2
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Answer:
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Step-by-step explanation:
Inverse Functions What is an Inverse Function? An inverse function is a function that will “undo” anything that the original function does. For example, we all have a way of tying our shoes, and how we tie our shoes could be called a function. So, what would be the inverse function of tying our shoes? The inverse function would be “untying” our shoes, because “untying” our shoes will “undo” the original function of tying our shoes. Let’s look at an inverse function from a mathematical point of view. Consider the function f(x) = 2x – 5. If we take any value of x and plug it into f(x) what would happen to that value of x? First, the value of x would get multiplied by 2 and then we would subtract 5. The two mathematical operations that are taking place in the function f(x) are multiplication and subtraction. Now let’s consider the inverse function. What two mathematical operations would be needed to “undo” f(x)? Division and addition would be needed to “undo” the multiplication and subtraction. A little farther down the page we will find the inverse of f(x) = 2x – 5, and hopefully the inverse function will contain both division and addition (see example 5). Notation If f(x) represents a function, then the notation 1 - f (x), inverse of f(x). Similarly, the notation - 1 g (x), read “f inverse of x”, will be used to denote the read “g inverse of x”, will be used to denote the inverse of g(x). Note: - 1 1 f (x). f (x) ¹ It is very important not to confuse function notation with negative exponents. Does the Function have an Inverse? Not all functions have an inverse, so it is important to determine whether or not a function has an inverse before we try and find the inverse. If a function does not have an inverse, then we need to realize the function does not have an inverse so we do not waste time trying to find something that does not exist. So how do we know if a function has an inverse? To determine if a function has an inverse function, we need to talk about a special type of function called a OnetoOne Function. A onetoone function is a function where each input (xvalue) has a unique output (yvalue). To put it another way, every time we plug in a value of x we will get a unique value of y, the same yvalue will never appear more than once. A onetoone function is special because only onetoone functions have an inverse function. Note: Only OnetoOne Functions have an inverse function. Examples – Now let’s look at a few examples to help demonstrate what a onetoone function is. Example 1: Determine if the function f = {(7, 3), (8, –5), (–2, 11), (–6, 4)} is a onetoone function. The function f is a onetoone function because each of the yvalues in the ordered pairs is unique; none of the yvalues appear more than once. Since the function f is a onetoone function, the function f must have an inverse.