For the function f(x)=(x-9)/(7-2x), find f^(-1)(x). Answer: f^(-1)(x)=

Replace with y: To find the inverse function, f^(-1)(x), we first replace f(x) with y for easier manipulation. So, we have y = (x - 9) / (7 - 2x).Swap x and y: Next, we swap x and y to find the inverse. This gives us x = (y - 9) / (7 - 2y).Solve for y: Now, we solve for y. Multiply both sides by (7 - 2y) to get rid of the fraction. x * (7 - 2y) = y - 9.Distribute x: Distribute x on the left side to get 7x - 2xy = y - 9.Get y terms together: To solve for y, we need to get all the y terms on one side. Add 2xy to both sides and add 9 to both sides to get 7x + 9 = y + 2xy.Factor out y: Factor out y on the right side to get 7x + 9 = y(1 + 2x).Isolate y: Now, divide both sides by (1 + 2x) to isolate y. y = (7x + 9) / (1 + 2x).Final Inverse Function: We have found the inverse function. Therefore, f^(-1)(x) = (7x + 9) / (1 + 2x).

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Algebra 2

Simplify variable expressions using properties

Full solution

Q. For the function

f(x)=\frac{x-9}{7-2 x}

, find

f^{-1}(x)

\newline

Answer:

f^{-1}(x)=

Replace with $y$ : To find the inverse function, $f^{-1}(x)$ , we first replace $f(x)$ with $y$ for easier manipulation. $\newline$ So, we have $y = \frac{x - 9}{7 - 2x}$ .
Swap $x$ and $y$ : Next, we swap $x$ and $y$ to find the inverse. This gives us $x = \frac{y - 9}{7 - 2y}$ .
Solve for $y$ : Now, we solve for $y$ . Multiply both sides by $(7 - 2y)$ to get rid of the fraction. $\newline$ $x \cdot (7 - 2y) = y - 9$ .
Distribute $x$ : Distribute $x$ on the left side to get $7x - 2xy = y - 9$ .
Get $y$ terms together: To solve for $y$ , we need to get all the $y$ terms on one side. Add $2xy$ to both sides and add $9$ to both sides to get $7x + 9 = y + 2xy$ .
Factor out $y$ : Factor out $y$ on the right side to get $7x + 9 = y(1 + 2x)$ .
Isolate $y$ : Now, divide both sides by $(1 + 2x)$ to isolate $y$ . $\newline$ $y = \frac{7x + 9}{1 + 2x}$ .
Final Inverse Function: We have found the inverse function. Therefore, $f^{-1}(x) = \frac{7x + 9}{1 + 2x}$ .