Question

1. $f(x) = |x| + 2$ 6. $f(x) = \frac{x^2}{x^3 - x}$

Explanation:

Since the problem is not specified (e.g., finding domain, range, analyzing the function, etc.), I'll assume a common task like finding the domain for both functions. Let's solve them one by one.

For \( f(x) = |x| + 2 \) (Problem 5)

Step1: Recall domain definition

The domain of a function is all real numbers \( x \) for which the function is defined. The absolute value function \( |x| \) is defined for all real numbers, and adding 2 doesn't restrict the domain.

Step2: Determine domain

Since \( |x| \) is defined for all \( x \in \mathbb{R} \), the function \( f(x) = |x| + 2 \) is defined for all real numbers.

Step1: Find where denominator is zero

The function is a rational function, so we need to find \( x \) values that make the denominator \( x^3 - x = 0 \), as division by zero is undefined.
Factor the denominator: \( x^3 - x = x(x^2 - 1) = x(x - 1)(x + 1) \) (using difference of squares \( a^2 - b^2=(a - b)(a + b) \) for \( x^2 - 1 \)).

Step2: Solve for \( x \) in denominator

Set \( x(x - 1)(x + 1) = 0 \). By zero - product property, \( x = 0 \) or \( x - 1 = 0 \) (i.e., \( x = 1 \)) or \( x + 1 = 0 \) (i.e., \( x=-1 \)).

Step3: Determine domain

The domain is all real numbers except \( x = - 1,0,1 \).

Answer:

The domain of \( f(x) = |x| + 2 \) is \( (-\infty, \infty) \) (all real numbers).

For \( f(x) = \frac{x^2}{x^3 - x} \) (Problem 6)