QuestionFind the largest one-to-one domain for and its inverse function equation.
Studdy Solution
STEP 1
Assumptions1. The function is
. We are looking for the largest domain over which the function is one-to-one.
3. We are also looking for the inverse of the function over that domain.
STEP 2
The function is defined for all real numbers. However, it is not one-to-one over the entire real line because it repeats its values for positive and negative values of x.
STEP 3
To find a domain over which the function is one-to-one, we can restrict the domain to non-negative real numbers. This is because the function is decreasing for .So, the largest domain over which the function is one-to-one is .
STEP 4
Now, we need to find the inverse of the function over the domain .The inverse of a function is found by swapping the roles of and in the equation and solving for .So, let's start by setting and swapping and :
STEP 5
To solve for , we first isolate the square root term by taking the reciprocal of both sides
STEP 6
Then, square both sides to eliminate the square root
STEP 7
Subtract1 from both sides to isolate :
STEP 8
Finally, take the square root of both sides to solve for . Note that we only take the positive square root because we restricted the domain to :
So, the inverse of the function over the domain is .
Was this helpful?