The problem asks to write a program or define the domain of each of the following functions: 1. $f(x) = x^2 - 2x + 1$

1. Problem Description

The problem asks to write a program or define the domain of each of the following functions:

1. $f(x) = x^2 - 2x + 1$

2. $g(x) = \frac{x-1}{x} + 2$

3. $h(x) = \sqrt{x^2 - 1} - \frac{1}{x}$

2. Solution Steps

1. For $f(x) = x^2 - 2x + 1$, the domain is all real numbers since it's a polynomial.

2. For $g(x) = \frac{x-1}{x} + 2$, the domain is all real numbers except $x = 0$ because the denominator cannot be zero.

$x \ne 0$

3. For $h(x) = \sqrt{x^2 - 1} - \frac{1}{x}$, we have two restrictions: the expression inside the square root must be non-negative, and the denominator of the fraction cannot be zero.

$x^2 - 1 \ge 0$

$x^2 \ge 1$

$|x| \ge 1$, which means $x \ge 1$ or $x \le -1$.

Also, $x \ne 0$. Since $x \ge 1$ or $x \le -1$, $x \ne 0$ is already satisfied.

Therefore, the domain is $x \in (-\infty, -1] \cup [1, \infty)$.

3. Final Answer

1. The domain of $f(x)$ is all real numbers, or $(-\infty, \infty)$.

2. The domain of $g(x)$ is all real numbers except 0, or $(-\infty, 0) \cup (0, \infty)$.

3. The domain of $h(x)$ is $(-\infty, -1] \cup [1, \infty)$.