Math Problem Statement

The natural domain of the function f1(x) = g(2x) is [-1, 3]. What is the natural domain of f2(x) = g(log2(x))?

Solution

We are given two functions and their domains:

  1. f1(x)=g(2x)f_1(x) = g(2x) with the natural domain [1,3][-1, 3].
  2. We are tasked with finding the natural domain of f2(x)=g(log2x)f_2(x) = g(\log_2{x}).

Solution:

For the function f1(x)=g(2x)f_1(x) = g(2x), we are told that the domain of g(2x)g(2x) is [1,3][-1, 3]. This implies that the argument of the function, 2x2x, lies within this interval: 12x3-1 \leq 2x \leq 3 Solving for xx, we divide the entire inequality by 2: 12x32-\frac{1}{2} \leq x \leq \frac{3}{2} Thus, x[12,32]x \in \left[ -\frac{1}{2}, \frac{3}{2} \right].

Now, for the function f2(x)=g(log2x)f_2(x) = g(\log_2{x}), the argument log2x\log_2{x} must also lie within the domain [1,3][-1, 3]. So, we have: 1log2x3-1 \leq \log_2{x} \leq 3 To solve this inequality, we rewrite it as an exponential: 21x232^{-1} \leq x \leq 2^3 which simplifies to: 12x8\frac{1}{2} \leq x \leq 8 Thus, the natural domain of f2(x)f_2(x) is [12,8]\left[ \frac{1}{2}, 8 \right].

Final Answer:

The natural domain of f2(x)=g(log2x)f_2(x) = g(\log_2{x}) is [12,8]\left[ \frac{1}{2}, 8 \right].

Would you like any additional details or have any questions?

Here are 5 related questions for further exploration:

  1. How does changing the base of the logarithm affect the domain of a function like g(logbx)g(\log_b{x})?
  2. What is the domain of g(3x)g(3x) if the domain of g(x)g(x) is [0,5][0, 5]?
  3. How does the transformation of functions g(kx)g(kx) affect the domain compared to g(x)g(x)?
  4. What happens to the domain of f2(x)=g(lnx)f_2(x) = g(\ln{x}) compared to f2(x)=g(log2x)f_2(x) = g(\log_2{x})?
  5. How does shifting the function g(x)g(x) by a constant affect its domain?

Tip: When working with logarithmic functions, remember that the argument of the logarithm must always be positive.

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Math Problem Analysis

Mathematical Concepts

Functions
Logarithms
Domain of Functions
Inequalities

Formulas

Domain of logarithmic functions
Logarithmic inequality
Base-2 logarithms

Theorems

Logarithmic function domain theorem
Basic properties of logarithms

Suitable Grade Level

Grades 11-12 or early college level