To determine the domain of the composition [tex]\((f \circ g)(x)\)[/tex], where [tex]\( f(x) = 2x \)[/tex] and [tex]\( g(x) = \frac{1}{x} \)[/tex], follow these steps:
1. Understand the functions individually:
- The function [tex]\( f(x) = 2x \)[/tex] is defined for all real numbers. This means its domain is all real numbers.
- The function [tex]\( g(x) = \frac{1}{x} \)[/tex] is defined for all real numbers except [tex]\( x = 0 \)[/tex], since division by zero is undefined.
2. Explore the composition [tex]\( f(g(x)) \)[/tex]:
- The composition [tex]\( (f \circ g)(x) = f(g(x)) \)[/tex] means we first apply [tex]\( g(x) \)[/tex] then apply [tex]\( f \)[/tex] to the result.
- We need [tex]\( g(x) \)[/tex] to be defined, so [tex]\( x \)[/tex] cannot be 0 (since [tex]\( g(0) \)[/tex] is undefined).
3. Evaluate [tex]\( f(g(x)) \)[/tex]:
- Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
(f \circ g)(x) = f\left(g(x)\right) = f\left(\frac{1}{x}\right) = 2 \cdot \frac{1}{x} = \frac{2}{x}
\][/tex]
- For [tex]\( f(g(x)) \)[/tex] to be defined, [tex]\( \frac{2}{x} \)[/tex] must be defined.
- Similar to [tex]\( g(x) \)[/tex], [tex]\( \frac{2}{x} \)[/tex] is undefined when [tex]\( x \)[/tex] is 0.
4. Determine the domain:
- Since [tex]\( g(x) \)[/tex] and [tex]\( \frac{2}{x} \)[/tex] are both undefined when [tex]\( x = 0 \)[/tex], the domain of [tex]\( (f \circ g)(x) \)[/tex] is all real numbers except [tex]\( x = 0 \)[/tex].
Thus, the domain of [tex]\( (f \circ g)(x) \)[/tex] is:
all real numbers except [tex]\( x = 0 \)[/tex].
