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QuestionFind the inverse of f(x)=−x−3−5f(x)=-\sqrt{x-3}-5 and state the domain of f−1(x)f^{-1}(x) in interval notation.

Studdy Solution

STEP 1

Assumptions1. The function given is f(x)=−x−3−5f(x)=-\sqrt{x-3}-5 . We need to find the inverse function, f−1(x)f^{-1}(x)3. We also need to specify the domain of f−1(x)f^{-1}(x) in interval notation

STEP 2

To find the inverse function, we first replace f(x)f(x) with yy.
y=−x−−5y=-\sqrt{x-}-5

STEP 3

Next, we swap xx and yy to find the inverse function.
x=−y−3−5x=-\sqrt{y-3}-5

STEP 4

We solve this equation for yy to find the inverse function.First, we isolate the square root term by adding to both sides.
x+=−y−3x+=-\sqrt{y-3}

STEP 5

Next, we square both sides of the equation to eliminate the square root.
(x+5)2=(−y−3)2(x+5)^2=(-\sqrt{y-3})^2

STEP 6

implify the equation.
x2+10x+25=y−3x^2+10x+25=y-3

STEP 7

Finally, add3 to both sides to solve for yy.
x2+10x+28=yx^2+10x+28=y

STEP 8

Now we have the inverse function, f−1(x)f^{-1}(x).
f−1(x)=x2+10x+28f^{-1}(x)=x^2+10x+28

STEP 9

Next, we need to find the domain of f−(x)f^{-}(x).The domain of a function is the set of all possible input values (x-values) which will produce a valid output.
For the function f−(x)=x2+x+28f^{-}(x)=x^2+x+28, the domain is all real numbers because there are no restrictions on the values of xx that can be substituted into the function.
So, the domain of f−(x)f^{-}(x) in interval notation is (−∞,∞)(-\infty, \infty).

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