



Key equations
Rational Function 
$f(x)=\frac{P(x)}{Q(x)}=\frac{{a}_{p}{x}^{p}+{a}_{p1}{x}^{p1}+\mathrm{...}+{a}_{1}x+{a}_{0}}{{b}_{q}{x}^{q}+{b}_{q1}{x}^{q1}+\mathrm{...}+{b}_{1}x+{b}_{0}},Q(x)\ne 0$ 
Key concepts
Section exercises
Verbal
What is the fundamental difference in the algebraic representation of a polynomial function and a rational function?
The rational function will be represented by a quotient of polynomial functions.
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If the graph of a rational function has a removable discontinuity, what must be true of the functional rule?
The numerator and denominator must have a common factor.
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Can a graph of a rational function have no
x intercepts? If so, how?
Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.
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Algebraic
For the following exercises, find the domain of the rational functions.
For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.
$f\left(x\right)=\frac{2}{5x+2}$
V.A. at
$\text{\hspace{0.17em}}x=\u2013\frac{2}{5};\text{\hspace{0.17em}}$ H.A. at
$\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals
$\text{\hspace{0.17em}}x\ne \u2013\frac{2}{5}$
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$f(x)=\frac{x}{{x}^{2}+5x36}$
V.A. at
$\text{\hspace{0.17em}}x=4,\u20139;\text{\hspace{0.17em}}$ H.A. at
$\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals
$\text{\hspace{0.17em}}x\ne 4,\u20139$
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$f(x)=\frac{3x4}{{x}^{3}16x}$
V.A. at
$\text{\hspace{0.17em}}x=0,4,4;\text{\hspace{0.17em}}$ H.A. at
$\text{\hspace{0.17em}}y=0;$ Domain is all reals
$\text{\hspace{0.17em}}x\ne 0,4,\u20134$
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$f(x)=\frac{x+5}{{x}^{2}25}$
V.A. at
$\text{\hspace{0.17em}}x=5;\text{\hspace{0.17em}}$ H.A. at
$\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ Domain is all reals
$\text{\hspace{0.17em}}x\ne 5,5$
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$f\left(x\right)=\frac{42x}{3x1}$
V.A. at
$\text{\hspace{0.17em}}x=\frac{1}{3};\text{\hspace{0.17em}}$ H.A. at
$\text{\hspace{0.17em}}y=\frac{2}{3};\text{\hspace{0.17em}}$ Domain is all reals
$\text{\hspace{0.17em}}x\ne \frac{1}{3}.$
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For the following exercises, find the
x  and
y intercepts for the functions.
For the following exercises, describe the local and end behavior of the functions.
Questions & Answers
can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {3,1,3} Q={3,21} and R= {2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x4/x^22x15 then
x is different from 5&3
Seid
All real x except 5 and  3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴xsin⁴x= cos²xsin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴xsin⁴x = cos²xsin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me
Add 2over ×+3 +y4 over 5
simplify (×+a)with square root of two ×root 2 all over a
multiply 1over ×y{(×y)(×+y)} over ×y
For the first question, I got (3y2)/15
Second one, I got Root 2
Third one, I got 1/(y to the fourth power)
I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=2
graph the following linear equation using intercepts method.
2x+y=4
Ashley
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x2
UriEl
y=mx+b
you were already given the 'm' and 'b'.
so..
y=3x2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line.
where m = the slope & b = where the line crosses the yaxis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
0=3x2
2=3x
x=3/2
then .
y=3/2X2
I think
Given
co ordinates for x
x=0,(2,0)
x=1,(1,1)
x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=b+_Гb2(4ac)
______________
2a
I've run into this:
x = r*cos(angle1 + angle2)
Which expands to:
x = r(cos(angle1)*cos(angle2)  sin(angle1)*sin(angle2))
The r value confuses me here, because distributing it makes:
(r*cos(angle2))(cos(angle1)  (r*sin(angle2))(sin(angle1))
How does this make sense? Why does the r distribute once
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis
vice versa with y=x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As
'f(x)=y'.
According to Google,
"The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :)
Thomas
GREAT ANSWER THOUGH!!!
Darius
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks.
"Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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