How do I find the inverse functions from abs(x^2 - 3)? WolframAlpha gives 4 expressions.
Yes, it does! And the reason is exactly what Prove It said- the function is not "one to one". If you were to graph it you would see that there are some horizontal lines that cross the graph 4 times- so 4 different values of x that give the same y value.
Specifically, is a parabola that crosses the x-axis at , goes down to (0, -3), back up to the x-axis at , and then up. Taking the absolute value then "folds" that portion below the x-axis up above it.
If , then so . Taking the absolute value of a positive number doesn't change it so, for , the function is . To find the inverse function, swap x and y and solve for y: so and . But in this case, y (which was x but got swapped) is less than or equal to -3. y is negative so .
If , then so . Taking the absolute value multiplies that by -1 so, for , . Again, we swap x and y and solve for y. so and . Because, here, x< 0 becomes y< 0, we must take the negative sign. .
If , we have the same situation as the previous case except that y is positive so we take the positive sign on the square root: .
If , we have the same situation as the first case except that y is positive so we take the positive sign on the square root: .