Show that the functions `f: R_*->R_*`defined by `f(x)=1/x` is one-one and onto. where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*.

Explanation

To show that the function \( f: \mathbb{R}_* \to \mathbb{R}_* \) defined by \( f(x) = \frac{1}{x} \) is one-one and onto, we will follow these steps:

### Step 1: Show that \( f \) is one-one
To prove that \( f \) is one-one, we need to show that if \( f(x_1) = f(x_2) \), then \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \).
2. This means \( \frac{1}{x_1} = \frac{1}{x_2} \).
3. Cross multiplying gives us \( x_2 = x_1 \).
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