If f is a real valued funtion of a real variable defined by:
x^2 - 3x + 2
x-> ------------
x^2 +3x + 2
using the maximal domain convention. Determine whether f is injective and/or surjective and find its range. If f is not inective, demonstrate this by providing two different x-values x1 and x2 which satisfy f(x1) = f(x2) if f is not surjective demonstrate this by providing a value of y which is a member of the reals and for which no value of x satisfies f(x) = y
To determine whether the function f(x) is injective (one-to-one) or surjective (onto), we need to analyze its properties. Here are the steps to follow:
Step 1: Find the domain of the function.
The function is defined as a ratio of two polynomials. To ensure that the denominator is not zero, we need to find the values of x that make x^2 + 3x + 2 = 0. Solving this quadratic equation, we find that x = -2 and x = -1. Therefore, the domain of f is all real numbers except -2 and -1.
Step 2: Determine if f(x) is injective.
A function is injective if every distinct element in the domain corresponds to a distinct element in the range. To check this, we need to find whether f(x1) = f(x2) implies x1 = x2 for any x1 and x2 in the domain.
Let's assume x1 and x2 are two distinct values in the domain of f. By evaluating f(x1) and f(x2), we can determine if they are equal.
f(x1) = (x1^2 - 3x1 + 2) / (x1^2 + 3x1 + 2)
f(x2) = (x2^2 - 3x2 + 2) / (x2^2 + 3x2 + 2)
From here, we need to check whether f(x1) = f(x2) implies x1 = x2.
Step 3: Determine if f(x) is surjective.
A function is surjective if for every y in the range, there exists an x in the domain such that f(x) = y.
To check this, we need to determine the range of the function. The range is all the possible values that f(x) can take. To find the range, we can analyze the behavior of the function as x approaches positive and negative infinity.
As x approaches positive infinity, f(x) approaches 1.
As x approaches negative infinity, f(x) approaches -1.
Therefore, the range of f(x) is the interval [-1, 1].
Now, using the steps provided, you can compute f(x1), f(x2), and analyze if f(x) is injective and surjective.