For the function $g(x) = 8|x| - 2$, let's find the domain and range:

1. Domain:

• The domain of $g(x)$ refers to all possible input values for $x$.
• Since the absolute value function $|x|$ is defined for all real numbers, the function $g(x) = 8|x| - 2$ can accept any real number $x$.

Thus, the domain is: $\boxed{(-\infty, \infty)}$

2. Range:

• The absolute value $|x|$ is always non-negative, so $8|x| \geq 0$.
• This means $g(x) = 8|x| - 2 \geq -2$ for all $x$, since the smallest value of $8|x|$ is 0 when $x = 0$.
• The minimum value of $g(x)$ occurs when $x = 0$, and in that case: $g(0) = 8|0| - 2 = -2$
• As $|x|$ increases (for either positive or negative values of $x$), $g(x)$ increases without bound.

Thus, the range is: $\boxed{[-2, \infty)}$

Would you like further details on how the range was derived or have any questions?

Related Questions:

1. How would the domain and range change if the function were $g(x) = 8x - 2$?
2. What is the graph of the function $g(x) = 8|x| - 2$ and how does it look?
3. What is the behavior of $g(x) = 8|x| - 2$ as $x \to \infty$?
4. How does modifying the coefficient of $|x|$ affect the range?
5. What is the value of $g(x)$ at $x = -2$ and $x = 2$?

Tip:

The absolute value function $|x|$ creates symmetry about the y-axis. This is important when analyzing functions like $g(x) = 8|x| - 2$.