Chapter 9.3
Inverse of Rational Functions
Inverse of Rational Functions
Find the inverse of f(x) algebraically.
f(x)=(x+9)/(x-8)
f^-1(x)=(8x+9)/(x-1)
Determine the Horizontal Asymptote of the function. Justify your answer.
f(x)=(x+9)/(x^2-9)
Scenario 1: The degree of the numerator is less than the degree of the denominator, thus the Horizontal asymptote is y = 0.
y= 0 is HA
Simplify. Record any excluded values.
(x^2-5x-14)/(x^2-49)
(x+2)/(x+7),x≠-7,7
Find the inverse of f(x) algebraically.
f(x)=(2x+5)/(x-3)
f^-1(x)=(3x+5)/(x-2)
Determine the Domain of the function. Justify your response.
f(x)=(x^2-5x-8)/(x^2+2x-24)
{x|x ≠ −6,4}
(−∞,-6)∪(-6,4)U(4,∞)
The domain of a rational function excludes all values that make the denominator of the original function equal to zero since it will make the equation undefined.
Dustin multiplied (x^2-4x-21)/(x^2-7x)*(6x^2)/(x-1) and determined (6(x+3))/(x-1) x≠ 0, 1, 7 represented the simplified expression. Check Dustin’s work by simplifying the expression in the space provided. Was Dustin correct or incorrect? Justify your response.
Dustin was incorrect because the factor of x in the denominator only cancels with ONE of the factors in the numerator. It does not cancel the entire.
((x-7)(x+3))/(x(x-7))*((6*x*x))/(x-1)
Find the inverse of the function
f(x)=(2x+3)/(x-5)
and state the asymptotes of the inverse.
f^-1(x)=(5x+3)/(x-2
x=2 is the Vertical Asymptotes
y=5 is the Horizontal Asymptotes
Determine whether or not the function has any holes. If the function does have holes identify their location.
f(x)=(x^2+5x-14)/(x^2+9x+14)
The hole is located
x=-7,(-7,9/5)
Rose determined that the expression (3-x)/(x^2-x-6) simplified to -1/(x+2) ,x≠-2,3.
Was rose Correct or Incorrect? Justify your response.
(3-x)/(x^2-x-6)=-(x-3)/((x+2)(x-3))=-1/(x+2),x≠-2,3
Rose simplified the expression correctly. She factored and canceled the appropriate factors, then listed all excluded values.
Find the inverse of: (there is technically two different answers for this one)
f(x)=(x-8)/(x+1)
f^-1(x)=(-x-8)/(x-1)
OR
f^-1(x)=(x+8)/(1-x)
Determine the veritcal asymptotes of the function. Justify your response.
f(x)=(2x^2+7x+5)/(x^2+6x+5)
x = -5
The function factors to f(x)=((2x+5)(x+1))/((x+1)(x+5)) .The factor (x+1) cancels, indicating the location of a hole. The remaining factor in the denominator, (x+5) indicates the location of the vertical asymptote.
Divide. Record any excluded values.
(x^2-81)/(x^2+5x+6)÷(x^2+5x-36)/(7x+21)
(7(x+9))/((x+2)(x-4),x≠ -9,-3,-2,4
What is the 3 step process for finding the inverse of the original function algebraically?
Switch x and y, solve for y, and rewrite using function notation.
Aiden says that x = 0 is defined in the domain of the function f(x)=(x-2)/(x^2-4) . Is Aiden correct? Justify your response.
The factored form of the f(x)=(x-2)/((x+2)(x-2)) .The domain of the function excludes the values of x=-2 and x = 2. Aiden is correct in stating that x = 0 is defined in the domain of the function because it is not excluded from the domain.
Divide. Record any excluded values.
(x^2-17x+66)/x^2÷(x^2-36)/(x^2-8x)
((x-11)(x-8))/(x(x+6)),x≠-6,0,6,8