PLEASE I NEED HELPPPPP.. i have many exercises like that, just want to know where to start thanks.
f(x)= x^3+2x^2-x-2 / x^2+x-6
find:
1. domain
2. symmetry or not
3. determine if f(x)is: a)in lowest trem b)proper/improper
4. x-intercepts, y-intercepts
5. zeros of f(x) and their multiplicity
6. vertical asymptotes
7. ablique/horizontal asymtotes
No problem! I'll help you with each of the steps. Here's how you can approach each question step-by-step:
1. Domain: To find the domain of a function, you need to determine the values of x for which the function is defined. In this case, the only restriction is when the denominator (x^2 + x - 6) equals zero, because division by zero is undefined. So, set the denominator equal to zero and solve for x. The factors of x^2 + x - 6 are (x + 3)(x - 2), so the solutions to x^2 + x - 6 = 0 are x = -3 and x = 2. Thus, the domain of the function is all real numbers except x = -3 and x = 2.
2. Symmetry: To determine if a function is symmetric or not, you need to check whether it is even, odd, or neither. For this, you can check whether f(-x) = f(x) (even symmetry) or f(-x) = -f(x) (odd symmetry). You can simplify the function by multiplying both the numerator and the denominator by the least common multiple of their terms. After simplification, if f(-x) = f(x) holds true, the function is even symmetric. If f(-x) = -f(x) holds true, the function is odd symmetric. If neither is true, the function has no symmetry.
3. Proper/Improper: A proper rational function is a function where the degree of the numerator is less than the degree of the denominator. An improper function is a function where the degree of the numerator is greater than or equal to the degree of the denominator. In this case, since the degree of the numerator (3) is greater than the degree of the denominator (2), the function f(x) is improper.
4. X-intercepts and Y-intercepts: To find the x-intercepts, set the numerator of f(x) equal to zero and solve for x. To find the y-intercept, set x equal to zero and evaluate f(x). The x-intercepts are the values of x where the function crosses the x-axis, and the y-intercept is the point where the function intersects the y-axis.
5. Zeros and Their Multiplicity: The zeros of a rational function are the values of x that make the function equal to zero. To find the zeros, set the numerator of f(x) equal to zero and solve for x. The multiplicity of the zero is the number of times the corresponding factor appears in the numerator.
6. Vertical Asymptotes: Vertical asymptotes occur when the value in the denominator approaches zero but never equals zero. In this case, find the values of x that make the denominator (x^2 + x - 6) equal to zero. These will be the vertical asymptotes of the function.
7. Oblique/Horizontal Asymptotes: To determine whether there are any oblique (slant) or horizontal asymptotes, you can check the degrees of the numerator and the denominator. If the degree of the numerator is less than the degree of the denominator, the function has a horizontal asymptote at y = 0. If the degree of the numerator is exactly one greater than the degree of the denominator, the function has an oblique asymptote.
I hope this helps you get started with your exercises! Let me know if you have any specific questions on any of the steps.