Let $f\left(x\right)=\frac{x}{2x-1}$  and $g\left(x\right)=\frac{x}{x-1}$ . Then the domain of the funtion h(x) = f(g(x)) + g(f(x)) is all real numbers except

The correct option is the one stating that the domain is all real numbers except:

$-1,\frac{1}{2},\text{and}1$

To find the domain of the function h(x) = f(g(x)) + g(f(x)), we need to ensure that both terms, f(g(x)) and g(f(x)), are mathematically defined. This means analyzing the domains of the inner functions and the composite functions.

Step 1: Find the restrictions for f(g(x))

For the composition f(g(x)) to be defined, two conditions must be met:

First, the inner function g(x) must be defined. The denominator of g(x) cannot be zero:

$x-1\ne 0$

$x\ne 1$

Second, the output of g(x) must fall within the domain of the outer function f(x). Looking at f(x), its denominator cannot be zero, which means its input cannot equal 1/2. Therefore, we must have:

$g\left(x\right)\ne \frac{1}{2}$

Substituting the expression for g(x):

$\frac{x}{x-1}\ne \frac{1}{2}$

Cross-multiplying gives us:

$2x\ne x-1$

$x\ne -1$

Thus, from f(g(x)), we know that x cannot be 1 or -1.

Step 2: Find the restrictions for g(f(x))

Similarly, for g(f(x)) to be defined:

First, the inner function f(x) must be defined. Its denominator cannot be zero:

$2x-1\ne 0$

$x\ne \frac{1}{2}$

Second, the output of f(x) must be a valid input for the outer function g(x). Because the denominator of g(x) becomes zero when its input is 1, we must restrict f(x) from equaling 1:

$f\left(x\right)\ne 1$

Substituting the expression for f(x):

$\frac{x}{2x-1}\ne 1$

Multiplying both sides by the denominator yields:

$x\ne 2x-1$

Subtracting x from both sides and moving the constant gives:

$x\ne 1$

Thus, from g(f(x)), we see that x cannot be 1/2 or 1.

Step 3: Combine all restrictions

Combining the excluded values from both parts, x cannot be equal to -1, 1/2, or 1. If x equals any of these values, at least one part of the function h(x) will be undefined.

Therefore, the domain of h(x) is all real numbers except -1, 1/2, and 1.