Question 1131458: Find the inverse of each function. Is the inverse a function?
y=x^2-3
y=(x-2)^3+1
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20849)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
A relation is a function if there are no x values with more than one y value; that means there is no vertical line that crosses the graph of the relation in more than one place: it passes the vertical line test.
The inverse of a function can be found by switching the x and y. That means the inverse of a function is a function if there are no y values with more than one x value. That means the inverse is a function if the function itself passes the HORIZONTAL line test.
Your first example is a quadratic equation; it is a function, with a graph that is a parabola. No parabola passes the horizontal line test; the inverse of a quadratic function is never a function.
Your second example is a cubic function. Some cubic functions have inverses that are functions; some do not.
To find the inverses of functions, one method is to switch the x and y in the equation of the function and solve for the new y. The answer from the other tutor did that.
Another way to find inverses for relatively simple functions is to use the idea that an inverse "gets you back where you started". That is, whatever the function does to the input variable, the inverse "undoes" it. And it undoes it by performing the opposite operations in the opposite order.
Your first example is y = x^2-3. What the function does to x is
(1) square it; and
(2) subtract 3
The inverse function has to do the opposite operations, in the reverse order:
(1) add 3; and
(2) take the square root
The other tutor gave the wrong answer here; when taking square roots when working with functions, you have to use both positive and negative results.
The inverse for your first example is then  OR 
With two different y values for each x value (except x=-3) you of course have a case where the inverse is not a function.
And again you know it is not going to be, because the function itself does not pass the horizontal line test.
For your second example, we can (try to) find the inverse in the same way. What the second example does to the input variable is
(1) subract 2;
(2) cube it; and
(3) add 1
The inverse has to do the opposite operations, in the reverse order:
(1) subtract 1;
(2) take the cube root; and
(3) add 2
The inverse of this function is then
Since some cubic functions have inverses and others do not, we need to look carefully at this to see if this inverse is a function. A careful graph with a graphing calculator will show that this inverse IS a function. You can also look at the expression for the inverse and see that there is no way of getting two different y values for one x value.
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