Question

Известно, что f(x) = (2 + x) / (x - 2). Найдите f(1/x)

Answer 1

$f(x)=\frac{2+x}{x-2}\\\\f(\frac{1}{x})=\frac{2+\frac{1}{x} }{\frac{1}{x} -2}=\frac{\frac{2x+1}{x} }{\frac{1-2x}{x}}=\frac{(2x+1)*x}{(1-2x)*x}=\frac{2x+1}{1-2x}$

Answer 2

To find the expression for $f\left(\frac{1}{x}\right)$, we'll replace every occurrence of $x$ in the function $f(x) = \frac{2 + x}{x - 2}$ with $\frac{1}{x}$. So, we get:

$f\left(\frac{1}{x}\right) = \frac{2 + \frac{1}{x}}{\frac{1}{x} - 2}$

Now, let's simplify this expression. To do that, we'll find a common denominator for the fractions in the numerator and the denominator:

$f\left(\frac{1}{x}\right) = \frac{\frac{2x + 1}{x}}{\frac{1 - 2x}{x}}$

Now, to divide by a fraction, we can multiply by its reciprocal:

$f\left(\frac{1}{x}\right) = \frac{2x + 1}{x} \cdot \frac{x}{1 - 2x}$

Now, we can cancel out the $x$ terms:

$f\left(\frac{1}{x}\right) = \frac{2x + 1}{1 - 2x}$

So, the expression for $f\left(\frac{1}{x}\right)$ is $\frac{2x + 1}{1 - 2x}$.