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Question 389525: For the function f(x)=(x^2+x-6)/(x^2+2x-8)
a) Find the factored form
b) Find any vertical asymptotes
c) Find any holes
d) Find the intercepts
e) Find the horizontal asymptote
Found 2 solutions by haileytucki, robertb:
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! A)(x^(2)+x-6)/(x^(2)+2x-8)
In this problem 3*-2=-6 and 3-2=1, so insert 3 as the right hand term of one factor and -2 as the right-hand term of the other factor.
((x+3)(x-2))/(x^(2)+2x-8)
In this problem 4*-2=-8 and 4-2=2, so insert 4 as the right hand term of one factor and -2 as the right-hand term of the other factor.
((x+3)(x-2))/((x+4)(x-2))
Reduce the expression by canceling out the common factor of (x-2) from the numerator and denominator.
((x+3)(x-2))/((x+4)(x-2))
Reduce the expression by canceling out the common factor of (x-2) from the numerator and denominator.
(x+3)/(x+4)
B)f(x)=(x^(2)+x-6)/(x^(2)+2x-8)
In this problem 3*-2=-6 and 3-2=1, so insert 3 as the right hand term of one factor and -2 as the right-hand term of the other factor.
f(x)=((x+3)(x-2))/(x^(2)+2x-8)
In this problem 4*-2=-8 and 4-2=2, so insert 4 as the right hand term of one factor and -2 as the right-hand term of the other factor.
f(x)=((x+3)(x-2))/((x+4)(x-2))
Reduce the expression by canceling out the common factor of (x-2) from the numerator and denominator.
f(x)=((x+3)(x-2))/((x+4)(x-2))
Reduce the expression by canceling out the common factor of (x-2) from the numerator and denominator.
f(x)=(x+3)/(x+4)
The domain of an expression is all real numbers except for the regions where the expression is undefined. This can occur where the denominator equals 0, a square root is less than 0, or a logarithm is less than or equal to 0. All of these are undefined and therefore are not part of the domain.
(x+4)=0
Solve the equation to find where the original expression is undefined.
x=-4
The domain of the rational expression is all real numbers except where the expression is undefined.
x$-4_(-I,-4) U (-4,I)
The vertical asymptotes are the values of x that are undefined in the function.
x=-4
A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches I.
L[x:I,(x+3)/(x+4)]
The value of L[x:I,((x+3))/((x+4))] is 1.
1
The horizontal asymptote is the value of y as x approaches I.
y=1
A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches -I.
L[x:-I,(x+3)/(x+4)]
The value of L[x:-I,((x+3))/((x+4))] is 1.
1
The horizontal asymptote is the value of y as x approaches -I.
y=1
Complete the polynomial division of the expression to determine if there is any remainder.
x+4,x+3,-x-4,Mx-1,1
Split the solution into the polynomial portion and the remainder.
1-(1)/(x+4)
Since there is no polynomial portion from the polynomial division, there are no oblique asymptotes.
No Oblique Aymptotes
This is the set of all asymptotes for f(x)=((x+3))/((x+4)).
Vertical Asymptote: x=-4_Horizontal Aysmptote:y=1_No Oblique Aysmptotes
C)Holes?? Specify which function please.
D)f(x)=(x^(2)+x-6)/(x^(2)+2x-8)
To find the x-intercept, substitute in 0 for y and solve for x.
(0)=(x^(2)+x-6)/(x^(2)+2x-8)
Since x is on the right-hand side of the equation, switch the sides so it is on the left-hand side of the equation.
(x^(2)+x-6)/(x^(2)+2x-8)=(0)
In this problem 3*-2=-6 and 3-2=1, so insert 3 as the right hand term of one factor and -2 as the right-hand term of the other factor.
((x+3)(x-2))/(x^(2)+2x-8)=(0)
In this problem 4*-2=-8 and 4-2=2, so insert 4 as the right hand term of one factor and -2 as the right-hand term of the other factor.
((x+3)(x-2))/((x+4)(x-2))=(0)
Remove the parentheses around the expression 0.
((x+3)(x-2))/((x+4)(x-2))=0
Reduce the expression by canceling out the common factor of (x-2) from the numerator and denominator.
((x+3)(x-2))/((x+4)(x-2))=0
Reduce the expression by canceling out the common factor of (x-2) from the numerator and denominator.
(x+3)/(x+4)=0
Find the LCD (least common denominator) of ((x+3))/((x+4))+0.
Least common denominator: (x+4)
Multiply each term in the equation by (x+4) in order to remove all the denominators from the equation.
(x+3)/(x+4)*(x+4)=0*(x+4)
Simplify the left-hand side of the equation by canceling the common factors.
x+3=0*(x+4)
Simplify the right-hand side of the equation by multiplying out all the terms.
x+3=0
Since 3 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 3 from both sides.
x=-3
To find the y-intercept, substitute in 0 for x and solve for y.
y=((0)^(2)+(0)-6)/((0)^(2)+2(0)-8)
Expand the exponent (2) to the expression.
y=((0^(2))+(0)-6)/((0)^(2)+2(0)-8)
Squaring a number is the same as multiplying the number by itself (0*0). In this case, 0 squared is 0.
y=((0)+(0)-6)/((0)^(2)+2(0)-8)
Remove the parentheses that are not needed from the expression.
y=(0+0-6)/((0)^(2)+2(0)-8)
Combine all similar expressions.
y=(-6)/((0)^(2)+2(0)-8)
Expand the exponent (2) to the expression.
y=(-6)/((0^(2))+2(0)-8)
Squaring a number is the same as multiplying the number by itself (0*0). In this case, 0 squared is 0.
y=(-6)/((0)+2(0)-8)
Multiply 2 by each term inside the parentheses.
y=(-6)/(0+0-8)
Combine all similar expressions.
y=(-6)/(-8)
Remove all extra parentheses from the expression.
y=-(-(6)/(8))
Reduce the expression -(6)/(8) by removing a factor of 2 from the numerator and denominator.
y=-(-(3)/(4))
Solve the equation.
y=(3)/(4)
These are the x and y intercepts of the equation y=((x^(2)+x-6))/((x^(2)+2x-8)).
x=-3, y=(3)/(4)
E)Already answered in (B)
Answer by robertb(5830)  (Show Source):
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