Solved on Oct 29, 2023

Find the inverse of x3\sqrt{x-3} and determine its domain, including any restrictions from the original function.

STEP 1

Assumptions1. The function is f(x)=x3f(x)=\sqrt{x-3} . We need to find the inverse of this function and its domain

STEP 2

To find the inverse of a function, we switch the roles of xx and yy. So, we write the function as y=xy=\sqrt{x-} and then swap xx and yy to get x=yx=\sqrt{y-}.
x=yx=\sqrt{y-}

STEP 3

Now, we need to solve this equation for yy to get the inverse function. To do this, we square both sides of the equation to eliminate the square root.
(x)2=(y3)2(x)^2=(\sqrt{y-3})^2

STEP 4

implify the equation to get yy alone on one side.
x2=y3x^2=y-3

STEP 5

Add3 to both sides to isolate yy.
y=x2+3y=x^2+3This is the inverse function, which we can write as f1(x)=x2+3f^{-1}(x)=x^2+3.

STEP 6

Now, we need to find the domain of the inverse function. The domain of a function is the set of all possible input values (x-values) that will give real output values (y-values).
For the inverse function f1(x)=x2+3f^{-1}(x)=x^2+3, the domain is all real numbers because any real number can be squared and then added to3 to give a real number.
However, we also need to consider any restrictions inherited from the original function f(x)=x3f(x)=\sqrt{x-3}.

STEP 7

The original function f(x)=x3f(x)=\sqrt{x-3} has a domain restriction. The expression under the square root, x3x-3, must be greater than or equal to0 because we cannot take the square root of a negative number.
So, x30x-3 \geq0.

STEP 8

olve this inequality for xx to find the domain restriction.
x3x \geq3

STEP 9

This means that the original function f(x)=x3f(x)=\sqrt{x-3} is defined for all xx greater than or equal to3. This restriction is inherited by the inverse function. So, the domain of the inverse function f(x)=x2+3f^{-}(x)=x^2+3 is all xx greater than or equal to3.
In conclusion, the inverse function of f(x)=x3f(x)=\sqrt{x-3} is f(x)=x2+3f^{-}(x)=x^2+3 and its domain is all xx greater than or equal to3.

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