what are the key features of x/(x^2-3x+2)?
how would describe it at x=0, and x=inf, and any asymptotes?
yes and domain and range pls.
describe it, I will critique.
To determine the key features of the function f(x) = x/(x^2 - 3x + 2), also known as a rational function, we need to examine its domain, intercepts, asymptotes, and behavior towards positive and negative infinity.
1. Domain: The domain of this function is all real numbers except those that make the denominator equal to zero. To find these values, we set the denominator equal to zero and solve for x: x^2 - 3x + 2 = 0. Factoring or using the quadratic formula, we find that x = 1 and x = 2. Thus, the domain is all real numbers except x = 1 and x = 2.
2. Intercepts: To find the x-intercept, we set the function equal to zero and solve for x: x/(x^2 - 3x + 2) = 0. Since a fraction is only equal to zero when the numerator is zero, then the x-intercept is x = 0. To find the y-intercept, we substitute x = 0 into the function: f(0) = 0/(0^2 - 3(0) + 2) = 0/2 = 0. Therefore, the y-intercept is 0.
3. Vertical Asymptotes: Vertical asymptotes occur when the function approaches infinity or negative infinity as x approaches a particular value. In this case, the function has vertical asymptotes when the denominator becomes zero. We already found that the denominator is zero when x = 1 and x = 2. As a result, the function has vertical asymptotes at x = 1 and x = 2.
4. Horizontal Asymptotes: To determine the horizontal asymptotes, we examine the behavior of the function as x approaches positive or negative infinity. To do this, we can analyze the degree of the numerator and denominator. In this case, the numerator is of degree 1 (x) and the denominator is of degree 2 (x^2 - 3x + 2). Since the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote at y = 0.
To summarize, the key features of the function f(x) = x/(x^2 - 3x + 2) are:
- Domain: All real numbers, except x = 1 and x = 2.
- Intercepts: x-intercept at x = 0, y-intercept at y = 0.
- Vertical Asymptotes: x = 1 and x = 2.
- Horizontal Asymptote: y = 0.