Okay, so I need to find the value of the inverse function f⁻¹ at 1/5 for the function f(x) = 2/(x+1). Hmm, let me start by recalling how inverse functions work. If f(x) maps x to y, then the inverse function f⁻¹(y) should map y back to x. So, basically, I need to find the x such that when I plug it into f(x), I get 1/5. 

Let me write down the equation: f(x) = 1/5. And since f(x) is given by 2/(x+1), I can set that equal to 1/5 and solve for x. That sounds like a plan. 

So, starting with the equation:

2/(x + 1) = 1/5

I need to solve for x. Let me cross-multiply to eliminate the fractions. Multiplying both sides by (x + 1) and 5 should work. Wait, maybe cross-multiplying is the right approach here. 

Cross-multiplying gives:

2 * 5 = 1 * (x + 1)

So, 10 = x + 1

Then, subtract 1 from both sides:

x = 10 - 1

x = 9

Wait, so x is 9? Let me check that. If I plug x = 9 into the original function f(x), do I get 1/5?

f(9) = 2/(9 + 1) = 2/10 = 1/5. Yep, that works. So the inverse function at 1/5 is 9. Therefore, f⁻¹(1/5) = 9. 

But just to make sure I didn't skip any steps or make a mistake, let me go through the process again. 

Starting with the equation:

2/(x + 1) = 1/5

To solve for x, first, multiply both sides by (x + 1):

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

Then, multiply both sides by 5 to eliminate the fraction:

10 = x + 1

Subtract 1:

x = 9

Same result. So that's correct. 

Alternatively, maybe another way to verify is to find the inverse function f⁻¹(y) in general and then plug in y = 1/5. Let's try that approach to cross-check. 

To find the inverse function, we start with the equation:

y = 2/(x + 1)

We need to solve for x in terms of y. Let's do that step by step.

Start with:

y = 2/(x + 1)

Multiply both sides by (x + 1):

y(x + 1) = 2

Then, divide both sides by y:

x + 1 = 2/y

Subtract 1:

x = (2/y) - 1

So, the inverse function is f⁻¹(y) = (2/y) - 1. 

Now, plugging in y = 1/5:

f⁻¹(1/5) = (2/(1/5)) - 1 = 2 * 5 - 1 = 10 - 1 = 9

Same answer. That confirms it. 

Wait a second, just to make sure I didn't make any arithmetic errors in the inverse function. Let me check the algebra again. 

Starting with y = 2/(x + 1). To solve for x, first, multiply both sides by (x + 1):

y(x + 1) = 2

Then, divide both sides by y:

x + 1 = 2/y

Subtract 1:

x = (2/y) - 1

Yes, that's correct. So when y is 1/5, substituting gives 2/(1/5) which is 10, minus 1 is 9. 

Alternatively, maybe I can use another method, like composing functions. If f and its inverse are supposed to undo each other, then f(f⁻¹(y)) should equal y and f⁻¹(f(x)) should equal x. Let's test that with y = 1/5. 

First, compute f(f⁻¹(1/5)). According to the inverse function we found, f⁻¹(1/5) is 9. Then f(9) is 2/(9 + 1) = 2/10 = 1/5, which is correct. 

Similarly, compute f⁻¹(f(x)). Let's take x = 9. Then f(9) = 1/5, and f⁻¹(1/5) = 9. So that works. 

Another check: Let's pick another value, say x = 0. Then f(0) = 2/(0 + 1) = 2. Then f⁻¹(2) should be 0. Using the inverse function: (2/2) - 1 = 1 - 1 = 0. Correct. 

Another example: x = 1. Then f(1) = 2/(1 + 1) = 1. Then f⁻¹(1) = (2/1) - 1 = 2 - 1 = 1. Correct. 

So all these checks work. Therefore, I can be confident that f⁻¹(1/5) is indeed 9. 

I think that's thorough. I considered multiple methods: solving the equation directly, finding the inverse function generally, and verifying with examples. All lead to the same result. So, unless I made a fundamental mistake in algebra (which I don't see), the answer should be 9.
Therefore, the value of $f^{-1}\left(\frac{1}{5}\right)$ is $\boxed{9}$.
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