QuestionFind the inverse of the function $f(x)=9+\sqrt{4x-9}$ for $x \geq 9$ .

Studdy Solution

STEP 1

Assumptions1. The function $f(x)$ is given as $f(x)=9+\sqrt{4x-9}$ . We are asked to find the inverse of the function $f(x)$ , denoted as $f^{-1}(x)$ 3. The domain of the inverse function is $x \geq9$

STEP 2

To find the inverse of a function, we first replace $f(x)$ with $y$ . This gives us $y =9+\sqrt{4x-9}$

STEP 3

Next, we swap $x$ and $y$ . This means we replace every $x$ with $y$ and every $y$ with $x$ . This gives us $x =9+\sqrt{y-9}$

STEP 4

Now we need to solve this equation for $y$ to find $f^{-1}(x)$ . We start by isolating the square root term. We do this by subtracting9 from both sides $x-9 = \sqrt{4y-9}$

STEP 5

Next, we square both sides of the equation to eliminate the square root $(x-9)^2 = (4y-9)$

STEP 6

Expanding the left side of the equation gives us $x^2 -18x +81 =4y -9$

STEP 7

Next, we isolate $y$ by adding9 to both sides and then dividing by4 $y = \frac{x^2 -18x +90}{4}$

STEP 8

Finally, we denote $y$ as $f^{-1}(x)$ to indicate that $y$ is the inverse of the function $f(x)$ $f^{-1}(x) = \frac{x^2 -18x +90}{4}$ So, the inverse of the function $f(x)=+\sqrt{4x-}$ is $f^{-1}(x) = \frac{x^2 -18x +90}{4}$ for $x \geq$ .

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