Question

question 4 of 10 step 1 of 2
00:52:03
consider the following function on the given domain.
$a(x) = (x + 1)^2 + 3, x \geq -1$
step 1 of 2: find a formula for the inverse of the function on the given domain, if possible.
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$a^{-1}(x) = $
$circ$ does not have an inverse fun

Explanation:

Step1: Replace $A(x)$ with $y$

$y = (x+1)^2 + 3, \ x \geq -1$

Step2: Swap $x$ and $y$

$x = (y+1)^2 + 3, \ y \geq -1$

Step3: Isolate the squared term

$x - 3 = (y+1)^2$

Step4: Solve for $y$ (take positive root, since $y \geq -1$ ensures $y+1 \geq 0$)

$y+1 = \sqrt{x - 3}$
$y = \sqrt{x - 3} - 1$

Step5: Replace $y$ with $A^{-1}(x)$

$A^{-1}(x) = \sqrt{x - 3} - 1$ (domain $x \geq 3$)

Answer:

$A^{-1}(x) = \sqrt{x - 3} - 1$