Does f(x)=-4x/x^2+4 have a vertical asymptote and why or why not?

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anonymous · Answer

f(x)=-(4x)/(x^(2))+4 Remove the common factors that were cancelled out. f(x)=-(4)/(x)+4 Cancel the common factor of x in -(4)/(x) since -(4)/(x)=((-4*x))/((x*x)). f(x)=-(4)/(x)+4 Remove the common factors that were cancelled out. f(x)=-(4)/(x)+4 To add fractions, the denominators must be equal. The denominators can be made equal by finding the least common denominator (LCD). In this case, the LCD is x. Next, multiply each fraction by a factor of 1 that will create the LCD in each of the fractions. f(x)=-(4)/(x)+4*(x)/(x) Multiply 4 by x to get 4x. f(x)=-(4)/(x)+(4x)/(x) Combine the numerators of all expressions that have common denominators. f(x)=(-4+4x)/(x) Reorder the polynomial -4+4x alphabetically from left to right, starting with the highest order term. f(x)=(4x-4)/(x) Factor out the GCF of 4 from the expression 4x. f(x)=(4(x)-4)/(x) Factor out the GCF of 4 from the expression -4. f(x)=(4(x)+4(-1))/(x) Factor out the GCF of 4 from 4x-4. f(x)=(4(x-1))/(x) 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=0 The domain of the rational expression is all real numbers except where the expression is undefined. x$0_(-I,0) U (0,I) The vertical asymptotes are the values of x that are undefined in the function. x=0 A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches I. L[x:I,(4(x-1))/(x)] Multiply 4 by each term inside the parentheses (x-1). L[x:I,(4(x-1))/(x)]=(4(x)+4(-1))/(x) Multiply 4 by the x inside the parentheses. L[x:I,(4(x-1))/(x)]=(4*x+4(-1))/(x) Multiply 4 by x to get 4x. L[x:I,(4(x-1))/(x)]=(4x+4(-1))/(x) Multiply 4 by the -1 inside the parentheses. L[x:I,(4(x-1))/(x)]=(4x-4*1)/(x) Multiply 4 by each term inside the parentheses. L[x:I,(4(x-1))/(x)]=(4x-4)/(x) Divide each term in the numerator by the denominator. L[x:I,(4(x-1))/(x)]=(4x)/(x)-(4)/(x) Simplify each expression in (4x)/(x)-(4)/(x). L[x:I,(4(x-1))/(x)]=4-(4)/(x) As x approaches I, x approaches I. L[x:I,(4(x-1))/(x)]=4-(4)/(I) The limit of -(4)/(x) as x approaches I is 0 L[x:I,(4(x-1))/(x)]=4+0 Add 0 to 4 to get 4. L[x:I,(4(x-1))/(x)]=4 The value of L[x:I,(4(x-1))/(x)] is 4. 4 The horizontal asymptote is the value of y as x approaches I. y=4 A horizontal asymptote can potentially be found by finding the limit of the function as the value approaches -I. L[x:-I,(4(x-1))/(x)] Multiply 4 by each term inside the parentheses (x-1). L[x:-I,(4(x-1))/(x)]=(4(x)+4(-1))/(x) Multiply 4 by the x inside the parentheses. L[x:-I,(4(x-1))/(x)]=(4*x+4(-1))/(x) Multiply 4 by x to get 4x. L[x:-I,(4(x-1))/(x)]=(4x+4(-1))/(x) Multiply 4 by the -1 inside the parentheses. L[x:-I,(4(x-1))/(x)]=(4x-4*1)/(x) Multiply 4 by each term inside the parentheses. L[x:-I,(4(x-1))/(x)]=(4x-4)/(x) Divide each term in the numerator by the denominator. L[x:-I,(4(x-1))/(x)]=(4x)/(x)-(4)/(x) Simplify each expression in (4x)/(x)-(4)/(x). L[x:-I,(4(x-1))/(x)]=4-(4)/(x) As x approaches -I, x approaches -I. L[x:-I,(4(x-1))/(x)]=4-(4)/(-I) The limit of -(4)/(x) as x approaches -I is 0 L[x:-I,(4(x-1))/(x)]=4+0 Add 0 to 4 to get 4. L[x:-I,(4(x-1))/(x)]=4 The value of L[x:-I,(4(x-1))/(x)] is 4. 4 The horizontal asymptote is the value of y as x approaches -I. y=4 Since there is no remainder from the polynomial division, there are no oblique asymptotes. No Oblique Aymptotes This is the set of all asymptotes for f(x)=(4(x-1))/(x). Vertical Asymptote: x=0_Horizontal Aysmptote:y=4_No Oblique Aysmptotes

anonymous · Answer

so: 
Vertical Asymptote: x=0
Horizontal Aysmptote:y=4
No Oblique Aysmptotes

anonymous · Answer

thank you!

anonymous · Answer

your welcome :)