Math  /  Algebra

Questionf(x)=2xe+2ex2ex+2ex2e f(x) = 2x^{e} + 2e^{x} - 2ex + \frac{2e}{x} - 2e

Studdy Solution

STEP 1

1. The function f(x)=2xe+2ex2ex+2ex2e f(x) = 2x^e + 2e^x - 2ex + \frac{2e}{x} - 2e involves the mathematical constant e e .
2. We are analyzing the function for its domain, range, or other properties unless specified.
3. The domain of the function is determined by the terms that could cause undefined behavior, such as division by zero.

STEP 2

1. Identify terms that affect the domain.
2. Determine the domain of the function.
3. Analyze the behavior of each term.
4. Simplify the function if possible.

STEP 3

Identify terms that affect the domain. In this function, the term 2ex\frac{2e}{x} involves division by xx, which means xx cannot be zero.

STEP 4

Determine the domain of the function. Since the only restriction comes from the term 2ex\frac{2e}{x}, the domain of f(x) f(x) is all real numbers except x=0 x = 0 .
Domain: xR,x0 \text{Domain: } x \in \mathbb{R}, x \neq 0

STEP 5

Analyze the behavior of each term: - 2xe2x^e is defined for all real xx. - 2ex2e^x is defined for all real xx. - 2ex-2ex is defined for all real xx. - 2ex\frac{2e}{x} is undefined at x=0x = 0. - 2e-2e is a constant term.

STEP 6

Simplify the function if possible. In this case, the function is already in its simplest form given the operations involved.
The domain of the function f(x)=2xe+2ex2ex+2ex2e f(x) = 2x^e + 2e^x - 2ex + \frac{2e}{x} - 2e is all real numbers except x=0 x = 0 .

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