Math Problem Statement

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)

Solution

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

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

To find the domain of g(x)g(x), we need to determine for which xx-values the argument of ff, i.e., 2x2x, lies within the domain of f(x)f(x).

The domain of f(x)f(x) is [12,22][-12, 22], so we require: 122x22-12 \leq 2x \leq 22

Dividing through by 2: 6x11-6 \leq x \leq 11

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

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

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

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

Conclusion

  • The domain of g(x)g(x) is [6,11][-6, 11].
  • The range of g(x)g(x) is [6,0][-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(x2)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)f(x) had a different domain, say [10,20][-10, 20], how would that change the domain of g(x)=f(2x)g(x) = f(2x)?
  4. How would the domain and range of f(x+3)f(x+3) compare to f(x)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)f(2x)), its domain shrinks, but the range remains unchanged.

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Math Problem Analysis

Mathematical Concepts

Functions
Transformations
Domain and Range

Formulas

Domain of g(x) = f(2x): -12 ≤ 2x ≤ 22, Range of g(x) remains same as f(x)

Theorems

Horizontal scaling transformations

Suitable Grade Level

Grades 10-12