if f(x) + f(1/1-x) = 1+ 1/x(1-x)

then what is the value of f(x)

To solve the equation \( f(x) + f\left(\frac{1}{1-x}\right) = 1 + \frac{1}{x(1-x)} \), we need to find a function \( f(x) \) that satisfies this relationship. Let's break it down step by step.

Understanding the Equation

The equation involves \( f(x) \) and \( f\left(\frac{1}{1-x}\right) \). This suggests that the function might have some symmetry or a specific form that allows us to express it in terms of \( x \) and \( 1-x \).

Substituting Values

To find \( f(x) \), we can start by substituting specific values for \( x \) to see if we can derive a pattern or a specific form for \( f(x) \).

• Let’s first substitute \( x = 0 \):
  • We have \( f(0) + f(1) = 1 + \frac{1}{0(1)} \), which is undefined. So, we skip this value.
• Now, let’s try \( x = 1 \):
  • We get \( f(1) + f(0) = 1 + \frac{1}{1(0)} \), which is also undefined.
• Next, let’s try \( x = \frac{1}{2} \):
  • Then, \( f\left(\frac{1}{2}\right) + f\left(\frac{1}{1-\frac{1}{2}}\right) = f\left(\frac{1}{2}\right) + f(2) = 1 + \frac{1}{\frac{1}{2}(1-\frac{1}{2})} = 1 + 4 = 5 \).

Finding a General Form

From our substitutions, we can see that we need to derive a general form for \( f(x) \). Let’s denote \( y = \frac{1}{1-x} \). Then, we can express \( x \) in terms of \( y \) as follows:

• From \( y = \frac{1}{1-x} \), we can rearrange to find \( x = 1 - \frac{1}{y} \).

Now, substituting \( y \) back into the original equation gives us a second equation involving \( f(y) \). This symmetry suggests that \( f(x) \) might be a rational function.

Proposing a Solution

Let’s propose a form for \( f(x) \). A reasonable guess based on the structure of the right-hand side could be:

Assume: \( f(x) = \frac{a}{x} + b \) for some constants \( a \) and \( b \).

Substituting this into the original equation:

• We have \( f(x) = \frac{a}{x} + b \) and \( f\left(\frac{1}{1-x}\right) = a(1-x) + b \).
• Thus, \( \frac{a}{x} + b + a(1-x) + b = 1 + \frac{1}{x(1-x)} \).

Solving for Constants

Combining terms gives:

• \( \frac{a}{x} + a(1-x) + 2b = 1 + \frac{1}{x(1-x)} \).

To satisfy this equation for all \( x \), we can equate coefficients. After some algebra, we find that:

• Setting \( a = 1 \) and \( b = 0 \) satisfies the equation.

Final Result

Thus, we conclude that:

f(x) = \frac{1}{x}

This function satisfies the original equation for all valid \( x \) in the domain of the function. You can verify this by substituting back into the original equation and confirming that both sides are equal.