I want to learn how to find the domain, holes, x-intercepts, y- intercepts, vertical asymptotes, and horizontal asymptotes of a rational function.
I know it is against the rules in jiskha to do the homework for the students but if I have been looking over my notes and I just don't get this.
example of a rational function from my problems
f(x)=-8x-16/x^2-x-12
Please someone help I have like 40 of these and I really want to learn. A link would help as well.
First you have to make sure you post algebraic expressions correctly.
Recall that multiplication and division have priority over addition and subtraction, so you expression would be interpreted as:
f(x)=(-8x) -(16/x^2) -x -12
which I do not believe is the intention.
An easy rule to remember is whenever you are transcribing an expression involving division or fractions, add parentheses around the numerator and the denominator if there is more than one term in each. So you rational expression would read:
f(x)=(-8x-16) / (x^2-x-12)
and it would be mathematically correct.
Do you have a textbook? If you don't, and if you are serious about learning calculus, borrow one from the library, or buy a used book for 1/4 of the price at this time of the year (even that is quite expensive, unfortunately).
Do a search on Google about functions, and read up about domain and range, the definition of a functions, the fundamental concepts required later on in Calculus. A link could be:
http://en.wikipedia.org/wiki/Function_%28mathematics%29
Assuming now that you know what a function is, f(x) is the notation, and x is the variable which has to lie within limits of values. These values together make the domain.
A rational function typically (but nt always) contains vertical asymptotes. This occurs when the value of x is such that the denominator becomes zero. So they are easy to spot. In the given function, there are two such points, x=x1, and x=x2. Can you spot them?
There is no horizontal asymptote in the given function f(x) because the degree of the numerator is less than that of the denominator. If they are of the same degree, there would be a horizontal asymptote.
Try to sketch the expressions of the numerator, the denominator, and the function itself on the same graph and you will soon recognize their relationships.
Keep up the good work.
Thank you so much!!
You're welcome!
Wait a minute...
I goofed about the horizontal asymptote.
When the degree of the numerator is lower than that of the denominator (in the case of f(x) above), there is a horizontal asymptote at y=0. It is important to take the limit of f(x) as x->±∞ and find out if the function approaches the asymptote from the side of +y or -y.
If the degrees are equal, then the horizontal asymptote is at y=Lim x->∞ of f(x), which is simply the quotient of the leading coefficients.
Certainly, I can help you understand how to find the domain, holes, x-intercepts, y-intercepts, vertical asymptotes, and horizontal asymptotes of a rational function.
To begin, let's break down each component step by step:
1. Finding the domain:
The domain of a rational function is the set of all real numbers for which the function is defined. In this case, the only value that would make the function undefined is if the denominator equals zero. So to find the domain, set the denominator equal to zero and solve for x. In your example, the denominator is x^2 - x - 12. Set this equal to zero: x^2 - x - 12 = 0. Solve this quadratic equation to find the values of x that make the function undefined. The solutions to this equation will give you the values to exclude from the domain.
2. Finding holes:
Holes in a rational function occur when factors in the numerator and denominator cancel each other out. To find the location of a hole, factor both the numerator and the denominator and simplify the expression. If any common factors remain, they indicate the coordinates of the hole.
3. Finding x-intercepts:
X-intercepts occur when the numerator of the rational function equals zero. To find the x-intercepts, set the numerator equal to zero and solve for x.
4. Finding y-intercepts:
Y-intercepts occur when x equals zero. Evaluate the function by substituting x = 0 into the expression and solve for y.
5. Finding vertical asymptotes:
Vertical asymptotes occur when the denominator of the rational function equals zero (excluding any values that might result in holes). To find the vertical asymptotes, set the denominator equal to zero and solve for x.
6. Finding horizontal asymptotes:
To find the horizontal asymptotes, you need to examine the degrees of the numerator and the denominator. There are three different cases to consider:
a) If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
b) If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of both the numerator and denominator.
c) If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, you have an oblique asymptote.
For a more detailed explanation and additional practice problems, you can visit the following link: https://www.purplemath.com/modules/rtnlfcn.htm
Remember, practice is key to mastering these concepts. Be sure to work through plenty of examples and check your answers.