To determine the inverse of the function [tex]\( f(x) = \frac{x}{x+3} \)[/tex], we need to follow these steps:
### Step 1: Substitute [tex]\( y \)[/tex] for [tex]\( f(x) \)[/tex]
Let [tex]\( y = \frac{x}{x+3} \)[/tex]. This rewrites the function in terms of [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[ y = \frac{x}{x+3} \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
We need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. First, multiply both sides of the equation by [tex]\( (x + 3) \)[/tex] to get rid of the denominator:
[tex]\[ y(x + 3) = x \][/tex]
Next, distribute [tex]\( y \)[/tex] on the left side:
[tex]\[ yx + 3y = x \][/tex]
Now, we want to collect all terms involving [tex]\( x \)[/tex] on one side of the equation. Subtract [tex]\( yx \)[/tex] from both sides:
[tex]\[ 3y = x - yx \][/tex]
Factor [tex]\( x \)[/tex] out on the right side:
[tex]\[ 3y = x(1 - y) \][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( (1 - y) \)[/tex]:
[tex]\[ x = \frac{3y}{1 - y} \][/tex]
### Step 3: Replace [tex]\( x \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] and [tex]\( y \)[/tex] with [tex]\( x \)[/tex]
The expression we obtained for [tex]\( x \)[/tex] represents [tex]\( f^{-1}(y) \)[/tex]. By convention, we replace [tex]\( y \)[/tex] with [tex]\( x \)[/tex] to write the inverse function formally:
[tex]\[ f^{-1}(x) = \frac{3x}{1 - x} \][/tex]
### Conclusion
The inverse function [tex]\( f^{-1}(x) \)[/tex] of the given function [tex]\( f(x) = \frac{x}{x+3} \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{3x}{1 - x} \][/tex]
Thus, the inverse of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{-3x}{x - 1} \][/tex]
