Check it on your calculator, and you'll see it is. Sometimes the instructions say if the function is not one-to-one, then don't find the inverse function because there's not one.
So, always check before wasting time trying to find the inverse function. Now, if you're supposed to find the inverse, regardless of whether it is a function or not, then go ahead. When solving equations, you can add the same thing to both sides, subtract the same thing from both sides, multiply both sides by the same non-zero thing, and divide both sides by the same non-zero thing and still get the same solution without worrying about having to check your answer.
You can also apply a one-to-one function to both sides of an equation without worrying about introducing extraneous solutions solutions that work after doing something that didn't work before. This is not necessarily true with functions that aren't one-to-one like the squaring function where you should always check answers after you square both sides of an equation.
With one-to-one functions, you won't be introducing any extraneous solutions. Talk about powerful. You don't appreciate it now, and the book doesn't deal with it properly until you get to chapter 4 and deal with logarithmic and exponential functions, and even then they don't make as big of deal out of it as it is.
Okay, let's try one now. Take my word for it that exp x is a one-to-one function and is the inverse of ln x. Wow - more cohesiveness.
The inverse of a function is found by taking the [2 nd ] function. Look at it for other things on the calculator. The square root is the inverse of the square. If you look at the three trigonometric keys [sin], [cos], and [tan], their inverses are all found by using the [2 nd ] key.
I'm telling you - it all fits together. Let's plot them both in terms of x Even though we write f -1 x , the "-1" is not an exponent or power :. Hide Ads About Ads. Inverse Functions An inverse function goes the other way! Example: continued Just make sure we don't use negative numbers. A function has to be "Bijective" to have an inverse. The inverse of f x is f -1 y We can find an inverse by reversing the "flow diagram" Or we can find an inverse by using Algebra: Put "y" for "f x ", and Solve for x We may need to restrict the domain for the function to have an inverse.
What is A Function? Injective, Surjective and Bijective Sets. No Inverse. If any horizontal line intersects the graph of f more than once, then f does not have an inverse. Definition : A function f is one-to-one if and only if f has an inverse. The graph of f is a line with slope 3, so it passes the horizontal line test and does have an inverse. There are two steps required to evaluate f at a number x. First we multiply x by 3, then we add 2.
Thinking of the inverse function as undoing what f did, we must undo these steps in reverse order. The steps required to evaluate f -1 are to first undo the adding of 2 by subtracting 2. Then we undo multiplication by 3 by dividing by 3. Step 2 often confuses students. We could omit step 2, and solve for x instead of y, but then we would end up with a formula in y instead of x. The formula would be the same, but the variable would be different. To avoid this we simply interchange the roles of x and y before we solve.
This is the function we worked with in Exercise 1. From its graph shown above we see that it does have an inverse.
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