A function g(x) has the following characteristics:
1) It is rational with a quadratic polynomial in both the numerator and the denominator.
2) It has a removable discontinuity at x=1 and the limg(x) as x approaches 1 is -1/9
3) The graph of g(x) has a vertical asymptote at x=5
4) The graph of g(x) has a single x-intercept at x=2
Use this information to find:
1. g(x)
2. g(0)
3. Limg(x) as x approaches infinity
To find the function g(x) with the given characteristics, we will analyze each characteristic and use them to build the function step by step.
1. Rational Function with Quadratic Polynomial:
Since g(x) is rational and has quadratic polynomials in both the numerator and denominator, we can express g(x) as g(x) = (ax^2 + bx + c) / (dx^2 + ex + f), where a, b, c, d, e, and f are constants that we need to determine.
2. Removable Discontinuity at x = 1:
We know that g(x) has a removable discontinuity at x = 1, which means the function is undefined at x = 1 but can be made continuous by assigning a value to g(1). We are also given that the limit of g(x) as x approaches 1 is -1/9.
To account for this removable discontinuity, we let g(1) = -1/9. This means that the factor (x - 1) exists in both the numerator and the denominator. The function becomes g(x) = (x - 1)(ax + b) / (x - 1)(dx + e).
Simplifying, we get g(x) = (ax + b) / (dx + e).
3. Vertical Asymptote at x = 5:
We know that the graph of g(x) has a vertical asymptote at x = 5. This means that the denominator (dx + e) must have a factor of (x - 5) to approach infinity as x approaches 5. Without loss of generality, we can assume d = 1 since dividing dx by d leads to the same function.
Now our function becomes g(x) = (ax + b) / (x + e)(x - 5).
4. Single x-intercept at x = 2:
We are given that the graph of g(x) has a single x-intercept at x = 2. This implies that the numerator (ax + b) must have a factor of (x - 2), leading to a root of x = 2 when the numerator is equal to zero.
Equating the numerator to zero, we have 2a + b = 0. Solving for b, we get b = -2a.
Plugging in b = -2a into our current expression, g(x) = (ax + b) / (x + e)(x - 5), we have g(x) = (ax - 2a) / (x + e)(x - 5).
Therefore, the function g(x) that satisfies all the given characteristics is g(x) = (ax - 2a) / (x + e)(x - 5).
To find the specific values of a and e, we need additional information or specific values for g(0) and the limit of g(x) as x approaches infinity.
2. g(0):
To find g(0), we substitute x = 0 into our function g(x). We have g(0) = (a(0) - 2a) / (0 + e)(0 - 5).
Simplifying, g(0) = (-2a) / (0 - 5e).
3. Limit of g(x) as x approaches infinity:
To find the limit of g(x) as x approaches infinity, we consider the behavior of the function as x becomes very large. Since g(x) is a rational function, the highest power terms in the numerator and denominator will dominate as x approaches infinity.
Examining our current function g(x) = (ax - 2a) / (x + e)(x - 5), we see that as x approaches infinity, the terms ax and x^2 become the dominant terms. Thus, the limit of g(x) as x approaches infinity will depend on the ratio of these dominant terms.
Please provide specific values for g(0) and any additional information or constraints to find the relevant constants and calculate the specific values of g(0) and the limit of g(x) as x approaches infinity.