Identifying the Type of Function \(f(x) = \frac{x + 1}{x + 2}\)

The given function is defined as \(f(x) = \frac{x + 1}{x + 2}\) for all \(x \in R - \{-2\}\). The domain of the function is all real numbers except \(-2\), and the codomain is the set of real numbers \(R\).

Let's examine the structure of the function \(f(x)\). The numerator, \(x + 1\), is a polynomial of degree 1. The denominator, \(x + 2\), is also a polynomial of degree 1.

Now let's consider the definitions of the types of functions provided in the options:

• Rational Function: A function that can be expressed as the ratio of two polynomials, \(P(x)\) and \(Q(x)\), i.e., \(f(x) = \frac{P(x)}{Q(x)}\), where \(Q(x)\) is not the zero polynomial. The domain of a rational function is typically all real numbers \(x\) such that \(Q(x) \neq 0\).
• Modulus Function: A function of the form \(f(x) = |x|\), which gives the absolute value of \(x\).
• Polynomial Function: A function that can be expressed in the form \(a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\), where \(a_i\) are constants and \(n\) is a non-negative integer.
• Signum Function: A function defined as \(sgn(x) = 1\) if \(x > 0\), \(sgn(x) = -1\) if \(x < 0\), and \(sgn(x) = 0\) if \(x = 0\).

Comparing the given function \(f(x) = \frac{x + 1}{x + 2}\) with these definitions, we see that it is a ratio of two polynomials, \(P(x) = x+1\) and \(Q(x) = x+2\). The denominator \(Q(x) = x+2\) is not the zero polynomial, and the domain correctly excludes the value \(x = -2\) where the denominator would be zero.

Therefore, the function \(f(x) = \frac{x + 1}{x + 2}\) is an example of a Rational Function.

The other options are not applicable:

• It is not a Modulus Function as it does not involve the absolute value of \(x\).
• It is not a Polynomial Function because it is a ratio of polynomials, not a simple sum of terms with non-negative integer powers of \(x\).
• It is not a Signum Function as its output is not restricted to 1, -1, or 0.

Based on the structure and definition, the given function is a rational function.