SOLUTION: Given the function t(x)=-sqrt(x+2)+1, find the inverse [t-1(x)]. I've gotten as far as y=-x^2+3. I don't know if this is correct, I don't know how to figure out if it is. I'm pa

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Question 203520: Given the function t(x)=-sqrt(x+2)+1, find the inverse [t-1(x)].
I've gotten as far as y=-x^2+3. I don't know if this is correct, I don't know how to figure out if it is. I'm particularly perplexed about how I deal with the negative sign in front of the sqrt in the original problem.
These inverse problems are just mystifying - I don't get it. Please help me by showing me how you solve the problem.
Thanks so much!
Julia
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
One way to think of a function is that it takes an x-value and, some way or another, it comes up with a y-value. The inverse of that function does the inverse: it takes the y-values from the function and, some way or another, it comes up with the x-value the function used. In other words the roles of the x and y are swapped. And this is exactly how we can figure out an inverse:
  1. Write the function, using "y" instead of "x":
  2. Swap the "x" and the "y". This changes the equation from the equation of the function to the equation of the inverse:
  3. Solve the inverse equation for y:
    Subtract -1 from both sides:

    Square both sides:


    Subtract 2 from both sides:
  4. Replace the "y" with (Note that Algebra.com's software incorrectly places a "*" between and ):

The way to check is feed f(x) as an input to its inverse. It should work out to be x:



Check! (Note how everything except the "x" cancels at the end.)
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