The domain of the function f(x) is [−12,22] and the range is [−6,0]. Using interval notation, find the domain and range of g(x)=f(2x)

Given the function \( f(x) \) has a domain of \( [-12, 22] \) and a range of \( [-6, 0] \), we need to find the domain and range of \( g(x) = f(2x) \).

### Domain of \( g(x) = f(2x) \)

To find the domain of \( g(x) \), we need to determine for which \( x \)-values the argument of \( f \), i.e., \( 2x \), lies within the domain of \( f(x) \).

The domain of \( f(x) \) is \( [-12, 22] \), so we require:
\[
-12 \leq 2x \leq 22
\]

Dividing through by 2:
\[
-6 \leq x \leq 11
\]

Thus, the domain of \( g(x) \) is \( [-6, 11] \).

### Range of \( g(x) = f(2x) \)

The range of \( g(x) \) is determined by the range of \( f(x) \), because \( g(x) \) is just a horizontal scaling of \( f(x) \) by a factor of 2. Since the range of \( f(x) \) is \( [-6, 0] \) and the transformation does not affect the output values of \( f \), the range of \( g(x) \) remains the same as that of \( f(x) \).

Thus, the range of \( g(x) \) is \( [-6, 0] \).

### Conclusion
- The domain of \( g(x) \) is \( [-6, 11] \).
- The range of \( g(x) \) is \( [-6, 0] \).

Would you like further details on how transformations affect functions, or do you have any other questions?

#### Related Questions:
1. What would happen to the domain and range if \( g(x) = f\left(\frac{x}{2}\right) \)?
2. How does multiplying the argument of a function by a constant affect its graph?
3. If \( f(x) \) had a different domain, say \( [-10, 20] \), how would that change the domain of \( g(x) = f(2x) \)?
4. How would the domain and range of \( f(x+3) \) compare to \( f(x) \)?
5. What is the effect of vertical stretching on the range of a function?

#### Tip:
When a function is horizontally compressed (as in \( f(2x) \)), its domain shrinks, but the range remains unchanged.

The domain of the function f(x) is [−12,22] and the range is [−6,0]. Using interval notation, find the domain and range of g(x)=f(2x).

Learn how to determine the domain and range of the function g(x) = f(2x) when the domain of f(x) is [-12, 22] and its range is [-6, 0]. This problem covers key concepts like function transformations and horizontal scaling. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation