Search for:. Use the graph of a function to graph its inverse Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Figure 7. Figure 9. Solution This is a one-to-one function, so we will be able to sketch an inverse. Licenses and Attributions. In the second case we did something similar. Note as well that these both agree with the formula for the compositions that we found in the previous section.
We get back out of the function evaluation the number that we originally plugged into the composition. So, just what is going on here? In some way we can think of these two functions as undoing what the other did to a number. Function pairs that exhibit this behavior are called inverse functions.
Before doing that however we should note that this definition of one-to-one is not really the mathematically correct definition of one-to-one. This can sometimes be done with functions. Showing that a function is one-to-one is often a tedious and difficult process. We did need to talk about one-to-one functions however since only one-to-one functions can be inverse functions.
For the two functions that we started off this section with we could write either of the following two sets of notation. Now, be careful with the notation for inverses. This is one of the more common mistakes that students make when first studying inverse functions. The process for finding the inverse of a function is a fairly simple one although there is a couple of steps that can on occasion be somewhat messy.
But this can be simplified. We can tell before we reflect the graph whether or not any vertical line will intersect more than once by looking at how horizontal lines intersect the original graph! 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.
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