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


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 \( g(x) = 8|x| - 2 \).

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

Discover the domain and range of the absolute value function g(x) = 8|x| − 2. This solution explains how to analyze the function's input values and output behavior, highlighting key concepts like the absolute value and linear transformations. Suitable for students in Grades 9-11.

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