What are the domain and range of *g*(*x*) = 8|*x*| − 2?]

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!

What are the domain and range of g(x) = 8|x| − 2?

Understand the domain and range of the function g(x) = 8|x| - 2. Explore how the absolute value affects the function, and see how to calculate the minimum and maximum values. Suitable for high school students learning about functions and transformations.

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