Determine if the function \( f(x) = \frac{5x^2 + 3x + 2}{x^2 + 4x + 4} \) is a proper or improper function.
Answer
The function is considered to be a proper function if the degree of the numerator is less than or equal to the degree of the denominator. Since the degrees of the numerator and the denominator are equal, therefore, \( f(x) = \frac{5x^2 + 3x + 2}{x^2 + 4x + 4} \) is a proper function.
Step 1 ：First, we look at the degrees of the polynomials in the numerator and the denominator.
Step 2 ：The degree of the polynomial in the numerator is 2, because the highest power of \(x\) in the numerator is 2.
Step 3 ：The degree of the polynomial in the denominator is also 2, because the highest power of \(x\) in the denominator is also 2.
Step 4 ：The function is considered to be a proper function if the degree of the numerator is less than or equal to the degree of the denominator. Since the degrees of the numerator and the denominator are equal, therefore, \( f(x) = \frac{5x^2 + 3x + 2}{x^2 + 4x + 4} \) is a proper function.
