For the function f(x)=(-1)/(9-x), find f^(-1)(x). Answer: f^(-1)(x)=

Rewrite function with y: To find the inverse function, f^(-1)(x), we need to switch the roles of x and y in the original function and then solve for y. Let's start by rewriting the function with y instead of f(x): y = (-1)/(9-x)Switch x and y: Now, switch x and y to find the inverse: x = (-1)/(9-y)Multiply by (9-y): Next, we solve for y. Start by multiplying both sides of the equation by (9-y) to get rid of the fraction: x * (9 - y) = -1Distribute x: Distribute x on the left side of the equation: 9x - xy = -1Isolate y: Now, we want to isolate y on one side of the equation. Add xy to both sides: 9x = xy - 1Add 1: Next, add 1 to both sides to isolate the terms with y on one side: 9x + 1 = xyFactor out y: Now, we need to factor out y on the right side of the equation: 9x + 1 = y(x)Divide by x: To solve for y, divide both sides by x: y = (9x + 1) / xInverse function: Finally, we have the inverse function: f^(-1)(x) = (9x + 1) / x

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Algebra 2

Simplify variable expressions using properties

Full solution

Q. For the function

f(x)=\frac{-1}{9-x}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Rewrite function with y: To find the inverse function, $f^{-1}(x)$ , we need to switch the roles of $x$ and $y$ in the original function and then solve for $y$ . Let's start by rewriting the function with $y$ instead of $f(x)$ : $\newline$ $y = \frac{-1}{9-x}$
Switch x and y: Now, switch x and y to find the inverse: $x = \frac{-1}{9-y}$
Multiply by $(9-y)$ : Next, we solve for $y$ . Start by multiplying both sides of the equation by $(9-y)$ to get rid of the fraction: $\newline$ $x \cdot (9 - y) = -1$
Distribute $x$ : Distribute $x$ on the left side of the equation: $\newline$ $9x - xy = -1$
Isolate $y$ : Now, we want to isolate $y$ on one side of the equation. Add $xy$ to both sides: $\newline$ $9x = xy - 1$
Add $1$ : Next, add $1$ to both sides to isolate the terms with $y$ on one side: $\newline$ $9x + 1 = xy$
Factor out y: Now, we need to factor out y on the right side of the equation: $\newline$ $9x + 1 = y(x)$
Divide by x: To solve for y, divide both sides by x: $\newline$ $y = \frac{9x + 1}{x}$
Inverse function: Finally, we have the inverse function: $f^{-1}(x) = \frac{9x + 1}{x}$