How do I find the inverse functions from abs(x^2 - 3)? WolframAlpha gives 4 expressions.

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- Nov 15th 2012, 04:44 AMStuck Maninverse functions
How do I find the inverse functions from abs(x^2 - 3)? WolframAlpha gives 4 expressions.

- Nov 15th 2012, 05:20 AMProve ItRe: inverse functions
This will not have an inverse function - it's not one-to-one... You could restrict the domain to get an inverse function though...

- Nov 15th 2012, 06:03 AMHallsofIvyRe: inverse functions
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, $\displaystyle y= x^2- 3$ is a parabola that crosses the x-axis at $\displaystyle (-\sqrt{3}, 0)$, goes down to (0, -3), back up to the x-axis at $\displaystyle (\sqrt{3}, 0)$, and then up. Taking the absolute value then "folds" that portion below the x-axis up above it.

If $\displaystyle x\le -\sqrt{3}$, then $\displaystyle x^2\ge 3$ so $\displaystyle x^2- 3\ge 0$. Taking the absolute value of a positive number doesn't change it so, for $\displaystyle x\le -\sqrt{3}$, the function is $\displaystyle y= x^2- 3$. To find the inverse function, swap x and y and solve for y: $\displaystyle x= y^2- 3$ so $\displaystyle y^2= x+ 3$ and $\displaystyle y= \pm\sqrt{x+ 3}$. But in this case, y (which**was**x but got swapped) is less than or equal to -3. y is negative so $\displaystyle y= -\sqrt{x+ 3}$.

If $\displaystyle -\sqrt{3}< x\le 0$, then $\displaystyle x^2< 3$ so $\displaystyle x^2- 3< 0$. Taking the absolute value multiplies that by -1 so, for $\displaystyle -\sqrt{3}< x< 0$, $\displaystyle y= -(x^2- 3)= 3- x^2$. Again, we swap x and y and solve for y. $\displaystyle x= 3- y^2]$ so $\displaystyle y^2= 3- x$ and $\displaystyle y= \pm\sqrt{3- x}$. Because, here, x< 0 becomes y< 0, we must take the negative sign. $\displaystyle y= -\sqrt{3- x}$.

If $\displaystyle 0< x\le \sqrt{3}$, we have the same situation as the previous case except that y is positive so we take the positive sign on the square root: $\displaystyle y= \sqrt{3- x}$.

If $\displaystyle \sqrt{3}< x$, we have the same situation as the first case except that y is positive so we take the positive sign on the square root: $\displaystyle y= \sqrt{x- 3}$.