To determine the domain and range of the function $g(x) = 8|x| - 2$, let's break it down:

1. Domain:

The domain refers to the set of all possible input values for $x$. Since the absolute value function $|x|$ is defined for all real numbers, and multiplying it by 8 and subtracting 2 does not introduce any restrictions, the domain of $g(x)$ is:

$\text{Domain of } g(x) = (-\infty, \infty)$

2. Range:

The range refers to the set of all possible output values of $g(x)$.

The absolute value function $|x|$ is always non-negative, meaning $|x| \geq 0$. Therefore, the expression $8|x|$ will always be at least 0 (when $x = 0$) and increases as $|x|$ increases. Specifically:

• When $x = 0$, $g(0) = 8(0) - 2 = -2$.
• As $|x|$ increases, $8|x| - 2$ becomes larger, which means $g(x)$ will continue to grow without bound.

Thus, the minimum value of $g(x)$ is $-2$, and there is no upper bound. Therefore, the range of $g(x)$ is:

$\text{Range of } g(x) = [-2, \infty)$

Summary:

• Domain: $(-\infty, \infty)$
• Range: $[-2, \infty)$

Would you like more details or have any questions? Here are some related questions to explore:

1. How would the domain and range change if the function was $g(x) = 8x - 2$ (without the absolute value)?
2. What is the significance of the absolute value in this function?
3. How does the transformation $g(x) = 8|x| - 2$ affect the graph of the basic absolute value function $f(x) = |x|$?
4. Can you find the vertex of the graph of $g(x) = 8|x| - 2$?
5. What happens to the range if the function was $g(x) = -8|x| - 2$?

Tip: Always check for any transformations in the function, such as absolute value or multiplication, which can affect both the range and shape of the graph!