Let the function f be defined by f(x)=9+3x7−2x, then f-1(x) is

The correct option is 7x−9/2x+3.

To find the inverse function, denoted as $f^{-1}\left(x\right)$, of the given function $f\left(x\right)$, we follow a systematic algebraic procedure.

The original function is given by:
$f\left(x\right)=\frac{9+3x}{7-2x}$

Step 1: Set the function equal to $y$.
We replace $f\left(x\right)$ with $y$ to work with standard variables:
$y=\frac{9+3x}{7-2x}$

Step 2: Swap $x$ and $y$.
To find the inverse function, we exchange the positions of $x$ and $y$:
$x=\frac{9+3y}{7-2y}$

Step 3: Solve the new equation for $y$.
First, we multiply both sides by the denominator $(7-2y)$ to eliminate the fraction:
$x(7-2y)=9+3y$

Next, we expand the left side of the equation:
$7x-2xy=9+3y$

To solve for $y$, we need to gather all terms containing $y$ on one side of the equation, and all other terms on the opposite side. Let's add $2xy$ to both sides and subtract $9$ from both sides:
$7x-9=3y+2xy$

Now, we factor out $y$ from the right-hand side:
$7x-9=y(3+2x)$

Which is equivalent to:
$7x-9=y(2x+3)$

Finally, we isolate $y$ by dividing both sides by $(2x+3)$:
$y=\frac{7x-92x+3}{}$

Therefore, the inverse function is:
$f^{-1}\left(x\right)=\frac{7x-9}{2x+3}$

This matches the correct option 7x−9/2x+3.