Problem
Question 3 of 11 , Step 2 of 2 $4 / 15$ Correct Consider the following function on the given domain. \[ P(x)=(x-5)^{4}-4, x \geq 5 \] Step 2 of 2 : Find the domain of the inverse function, $P^{-1}(x)$. Express your answer as an inequality. Answer
Solution
Step 1 ：Consider the function \(P(x)=(x-5)^{4}-4\), where \(x \geq 5\).
Step 2 ：The domain of the inverse function, \(P^{-1}(x)\), is the range of the original function, \(P(x)\).
Step 3 ：Since \(P(x)\) is a fourth degree polynomial, it will have a minimum value at \(x=5\).
Step 4 ：The minimum value of \(P(x)\) is \(P(5) = (5-5)^4 - 4 = -4\).
Step 5 ：Since \(x \geq 5\), the function \(P(x)\) will increase without bound as \(x\) increases.
Step 6 ：Therefore, the range of \(P(x)\), and thus the domain of \(P^{-1}(x)\), is \([-4, \infty)\).
Step 7 ：\(\boxed{\text{The domain of the inverse function, } P^{-1}(x) \text{, is } [-4, \infty)}\)
