let f be the function defined by f x 0 if x is rational and f(x) = x if x is irrational and g(x) = 0 of x is irrational and g(x) = x if x is rational then the function (f-g)x is

To analyze the function defined by \( (f - g)(x) \), we first need to understand the individual functions \( f(x) \) and \( g(x) \). The function \( f \) is defined such that \( f(x) = 0 \) when \( x \) is rational and \( f(x) = x \) when \( x \) is irrational. Conversely, the function \( g \) is defined as \( g(x) = 0 \) when \( x \) is irrational and \( g(x) = x \) when \( x \) is rational. Let's break this down step by step.

Understanding the Functions

We can summarize the definitions of \( f \) and \( g \) as follows:

• For \( f(x) \):
  • If \( x \) is rational, then \( f(x) = 0 \).
  • If \( x \) is irrational, then \( f(x) = x \).
• For \( g(x) \):
  • If \( x \) is irrational, then \( g(x) = 0 \).
  • If \( x \) is rational, then \( g(x) = x \).

Calculating \( (f - g)(x) \)

Now, let's compute \( (f - g)(x) \) for both rational and irrational values of \( x \).

Case 1: \( x \) is Rational

When \( x \) is rational:

• From \( f(x) \), we have \( f(x) = 0 \).
• From \( g(x) \), we have \( g(x) = x \).

Thus, we can calculate:

\( (f - g)(x) = f(x) - g(x) = 0 - x = -x \).

Case 2: \( x \) is Irrational

When \( x \) is irrational:

• From \( f(x) \), we have \( f(x) = x \).
• From \( g(x) \), we have \( g(x) = 0 \).

Therefore, we find:

\( (f - g)(x) = f(x) - g(x) = x - 0 = x \).

Final Result

Combining both cases, we can express \( (f - g)(x) \) as follows:

• If \( x \) is rational, then \( (f - g)(x) = -x \).
• If \( x \) is irrational, then \( (f - g)(x) = x \).

In summary, the function \( (f - g)(x) \) can be succinctly defined as:

\( (f - g)(x) = \begin{cases} -x & \text{if } x \text{ is rational} \\ x & \text{if } x \text{ is irrational} \end{cases} \)

This function exhibits interesting behavior based on the nature of the input \( x \), showcasing how rational and irrational numbers interact within the framework of these defined functions.