Studdy AI Logo
Math

QuestionFind the inverse of f(x)=−x+9f(x)=-\sqrt{x+9} and specify the domain of f−1(x)f^{-1}(x) in interval notation. f−1(x)= f^{-1}(x)=

Studdy Solution

STEP 1

Assumptions1. The function given is f(x)=−x+9f(x)=-\sqrt{x+9}. . We are required to find the inverse of this function, denoted as f−1(x)f^{-1}(x).
3. We are also required to specify the domain of f−1(x)f^{-1}(x) in interval notation.

STEP 2

To find the inverse of a function, we first replace f(x)f(x) with yy.
y=−x+9y = -\sqrt{x+9}

STEP 3

Next, we swap xx and yy to find the inverse.
x=−y+9x = -\sqrt{y+9}

STEP 4

Now, we solve for yy to find the inverse function. First, we remove the negative sign by multiplying both sides by -1.
−x=y+9-x = \sqrt{y+9}

STEP 5

Next, we square both sides to remove the square root.
(−x)2=(y+9)2(-x)^2 = (\sqrt{y+9})^2

STEP 6

implify the equation.
x2=y+9x^2 = y+9

STEP 7

Finally, subtract9 from both sides to solve for yy.
x2−9=yx^2 -9 = y

STEP 8

Replace yy with f−1(x)f^{-1}(x) to get the inverse function.
f−1(x)=x2−f^{-1}(x) = x^2 -

STEP 9

Now, 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) that will give real output values.

STEP 10

For the function f−(x)=x2−9f^{-}(x) = x^2 -9, the domain is all real numbers because any real number can be squared and then subtracted by9 to give a real number.
Therefore, the domain of f−(x)f^{-}(x) in interval notation is (−∞,+∞)(-\infty, +\infty).
So, the inverse function of f(x)=−x+9f(x)=-\sqrt{x+9} is f−(x)=x2−9f^{-}(x) = x^2 -9 and the domain of f−(x)f^{-}(x) is (−∞,+∞)(-\infty, +\infty).

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord