QuestionFind a formula for the inverse of the function.
Studdy Solution
STEP 1
1. The function is defined for .
2. The inverse function, , will satisfy .
3. We need to solve for in terms of where .
STEP 2
1. Set up the equation for the inverse.
2. Solve for in terms of .
3. Verify the inverse function.
STEP 3
Set up the equation for the inverse by letting . Thus, we have:
STEP 4
To solve for in terms of , first square both sides to eliminate the square root:
Rearrange the equation to form a quadratic equation:
STEP 5
Solve the quadratic equation using the quadratic formula:
Here, , , and . Substitute these values into the formula:
Since , we choose the positive root:
STEP 6
Verify the inverse function by checking if . Substitute into :
Simplify the expression:
Thus, the inverse function is verified.
The formula for the inverse function is .
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