<Question>
If $f(x)=\dfrac{2}{x+1}$, then what is the value of $f^{-1}\left(\frac{1}{5}\right)$?
</Question>

<think>
<Plan_1>
Okay, so I need to find the value of the inverse function of f at 1/5, given that f(x) is 2 divided by (x + 1). Hmm, let's start by recalling what an inverse function does. The inverse function essentially reverses the operation of the original function. So, if f(x) maps x to y, then f inverse of y should map back to x. Therefore, to find f inverse of 1/5, I need to find the x value such that when I plug it into f(x), I get 1/5. Alternatively, another method is to find the inverse function formula first and then plug in 1/5 into that inverse function.

Let me try both approaches to see if I can figure it out.

First, let's try the algebraic method of finding the inverse function. To find the inverse of f(x), I need to solve the equation y = 2/(x + 1) for x in terms of y. Then, that expression for x will be f inverse(y). Let's do that step by step.

Starting with y = 2/(x + 1). I want to solve for x. Let's multiply both sides by (x + 1) to get rid of the denominator.
That gives y(x + 1) = 2. Then, distribute y: yx + y = 2. Now, isolate the terms with x. Subtract y from both sides: yx = 2 - y. Then, divide both sides by y to solve for x: x = (2 - y)/y.

So, that gives x in terms of y, which is (2 - y)/y. Therefore, the inverse function f inverse(y) is (2 - y)/y.
Let me check that.
If I substitute y = 1/5 into this inverse function, I should get the answer.

Calculating (2 - 1/5)/(1/5) = (10/5 - 1/5)/(1/5) = (9/5)/(1/5) = 9/5 * 5/1 = 9. So, that suggests that f inverse(1/5) is 9. Let me verify this by plugging back into the original function.

<prompt_1.1> Verify the result by plugging the x value back into the original function. </prompt_1.1>

Alternatively, maybe I should have done the steps more carefully. Let me go through the inverse function derivation again to make sure there's no mistake.

<prompt_1.2> Re-derive the inverse function to ensure its correctness. </prompt_1.2>
-----

Alternatively, if I didn't want to find the inverse function explicitly, I could solve the equation f(x) = 1/5 directly. Let's try that approach as a check.

<prompt_1.3> Solve the equation f(x) = 1/5 directly to verify the result. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
If x is 9, then f(9) = 2/(9 + 1) = 2/10 = 1/5. That works.
</execution_1.1>

<execution_1.2>
Starting with y = 2/(x + 1). Multiply both sides by (x + 1): y(x + 1) = 2. Then, x + 1 = 2/y. Subtract 1 from both sides: x = (2/y) - 1. Then, combining the terms: x = (2 - y)/y. Yes, that's the same as before. So, f inverse(y) = (2 - y)/y.
</execution_1.2>

<execution_1.3>
Given f(x) = 2/(x + 1), set that equal to 1/5:

2/(x + 1) = 1/5.

Cross-multiplying gives 2*5 = 1*(x + 1), so 10 = x + 1. Subtract 1 from both sides: x = 10 - 1 = 9.
</execution_1.3>
</Execution_1>

<Plan_2>
Based on execution_1.2:
Then, plugging y = 1/5 into that: (2 - 1/5)/(1/5).
Calculating the numerator first: 2 is 10/5, so 10/5 - 1/5 = 9/5. Then, dividing by 1/5 is the same as multiplying by 5, so 9/5 * 5 = 9.
-----

Based on execution_1.1, execution_1.2, execution_1.3:
Therefore, all methods lead to the same answer, x = 9. So, I can be confident that the value of f inverse(1/5) is 9.

<prompt_2.1> Conclude the solution by confirming the final answer using all methods. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Just to recap, the key steps were either solving for x in terms of y in the equation y = 2/(x + 1), which gives the inverse function, then substituting y = 1/5, or directly solving the equation 2/(x + 1) = 1/5 for x. Both approaches confirm that x = 9. Plugging x = 9 back into the original function gives 1/5, which is consistent. So, everything checks out. I don't see any mistakes in the reasoning here.
</execution_2.1>
</Execution_2>

<Final_answer>
Therefore, the value of $f^{-1}\left(\frac{1}{5}\right)$ is $\boxed{9}$.
</Final_answer>
</think>