To find the domain of each function from the given set, we need to analyze each function to identify any restrictions (such as division by zero or square roots of negative numbers). Let’s go through each function step by step.

1) $f(x) = 2x - 1$

This is a linear function, which is defined for all real values of $x$.

• Domain: $\mathbb{R}$ (All real numbers)

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

This is also a linear function with a constant denominator of 2, which is non-zero for all $x$.

• Domain: $\mathbb{R}$ (All real numbers)

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

This function involves a division, so we must exclude values of $x$ that make the denominator zero.

• The denominator becomes zero when $x + 1 = 0$, or $x = -1$.
• Domain: $x \in \mathbb{R}, x \neq -1$

4) $f(x) = \frac{1}{1 - x^2}$

This is another rational function. We need to exclude values of $x$ that make the denominator zero:

• $1 - x^2 = 0$
• $x^2 = 1$
• $x = \pm 1$
• Domain: $x \in \mathbb{R}, x \neq 1, x \neq -1$

5) $f(x) = \frac{1}{x^2 - x - 6}$

We must exclude values of $x$ that make the denominator zero. First, factor the quadratic:

• $x^2 - x - 6 = (x - 3)(x + 2)$
• The denominator becomes zero when $x - 3 = 0$ or $x + 2 = 0$, which gives $x = 3$ and $x = -2$.
• Domain: $x \in \mathbb{R}, x \neq 3, x \neq -2$

6) $f(x) = \sqrt{x - 2}$

For a square root function, the expression inside the square root must be non-negative:

• $x - 2 \geq 0$
• $x \geq 2$
• Domain: $x \in [2, \infty)$

Summary of Domains:

1. $f(x) = 2x - 1$ → $\mathbb{R}$
2. $f(x) = \frac{x + 1}{2}$ → $\mathbb{R}$
3. $f(x) = \frac{2}{x + 1}$ → $x \in \mathbb{R}, x \neq -1$
4. $f(x) = \frac{1}{1 - x^2}$ → $x \in \mathbb{R}, x \neq 1, x \neq -1$
5. $f(x) = \frac{1}{x^2 - x - 6}$ → $x \in \mathbb{R}, x \neq 3, x \neq -2$
6. $f(x) = \sqrt{x - 2}$ → $x \in [2, \infty)$

Would you like a more detailed explanation of any of these steps or have any questions?

Related Questions:

1. What are common techniques to find the domain of rational functions?
2. How can the domain of a square root function be determined?
3. What is the difference between a linear and rational function in terms of domain?
4. How can factoring help find the domain of functions with quadratics in the denominator?
5. What are some key restrictions to watch for when determining the domain of a function?

Tip:

Always check for any square roots or division in a function—these typically cause domain restrictions.