The domain of the function f(x) is open square bracket, minus, 8, comma, 1, close square bracket[−8,1] and the range is open square bracket, 1, comma, 17, close square bracket[1,17]. Using interval notation, find the domain and range of g, of, x, equals, f, left parenthesis, one quarter, x, right parenthesisg(x)=f( 
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 x).

Given the function \( g(x) = f\left(\frac{1}{4}x\right) \), we need to find the domain and range of \( g(x) \) based on the given domain and range of \( f(x) \).

### Domain of \( g(x) \):
The domain of \( f(x) \) is \([-8, 1]\), meaning that \( f(x) \) is only defined for values of \( x \) in this interval. For \( g(x) = f\left(\frac{1}{4}x\right) \), we need to figure out what values of \( x \) will result in inputs to \( f(x) \) that lie within the interval \([-8, 1]\).

1. To find the domain of \( g(x) \), solve the inequality:
   \[
   -8 \leq \frac{1}{4}x \leq 1
   \]
2. Multiply the entire inequality by 4 to eliminate the fraction:
   \[
   -32 \leq x \leq 4
   \]

Thus, the domain of \( g(x) \) is \([-32, 4]\).

### Range of \( g(x) \):
The range of \( f(x) \) is \([1, 17]\), and since \( g(x) \) is simply a transformation of \( f(x) \) via scaling, it does not affect the range. Hence, the range of \( g(x) \) remains the same as that of \( f(x) \), which is \([1, 17]\).

### Conclusion:
- Domain of \( g(x) \): \([-32, 4]\)
- Range of \( g(x) \): \([1, 17]\)

Would you like further details or have any questions?

Here are 5 related questions to expand on this:
1. What happens to the domain if the function was \( g(x) = f(2x) \) instead of \( f\left(\frac{1}{4}x\right) \)?
2. How does a horizontal compression or stretch affect the domain of a function?
3. What if \( f(x) \) had a different range? How would it impact \( g(x) \)'s range?
4. Can you visualize what transformations are occurring to the graph of \( f(x) \) to get \( g(x) \)?
5. How would the domain change if we had \( f\left(\frac{1}{k}x\right) \) for any constant \( k \)?

**Tip:** When transforming functions, pay close attention to how scaling factors inside the function affect the domain, as they cause horizontal stretches or compressions.

The domain of the function f(x) is [-8, 1] and the range is [1, 17]. Using interval notation, find the domain and range of g(x) = f(1/4x).

Learn how to find the domain and range of g(x) = f(1/4x) given the domain [-8, 1] and range [1, 17] of f(x). This problem covers key concepts in function transformations, interval notation, and properties of functions. Suitable for high school students in Grades 10-12.

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