Inverse Functions horizontal line test Recall: To determine if a functions inverse is also a function use the ________________________ For each graph below, determine whether or not the inverse would represent a function horizontal line test YES or NO YES or NO YES or NO YES or NO

Inverse Functions Every restrict the domain quadratic, absolute value _______ function has an inverse, but not every inverse will represent a function For the inverse of a function to be a function, it may be necessary to ___________________ Think about the parent functions we have worked with this semester. Which function(s) would never pass the horizontal line test? Both quadratic and absolute value functions will have inverse functions that exist on a restricted domain. To identify the domain, identify the _______ of the equation. restrict the domain quadratic, absolute value vertex

Vertex at _______________ Inverse function will exist on the domain: The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function. Vertex at _______________ Inverse function will exist on the domain: __________ or __________ (−3,−4) 𝑥≤−3 𝑥≥−3

Vertex at _______________ Inverse function will exist on the domain: The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function. Vertex at _______________ Inverse function will exist on the domain: __________ or __________ (3,−2) 𝑥≤3 𝑥≥3

Vertex at _______________ Inverse function will exist on the domain: The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function. Vertex at _______________ Inverse function will exist on the domain: __________ or __________ (1,−3) 𝑥≤1 𝑥≥1

Vertex at _______________ Inverse function will exist on the domain: The function below does not pass the horizontal line test. Identify the vertex and use this point to restrict the domain so that the inverse will represent a function. Vertex at _______________ Inverse function will exist on the domain: __________ or __________ (−1,4) 𝑥≤−1 𝑥≥−1

Restricted Domain for Inverse of Quadratics As we just saw, when finding the restricted domain for the inverse of a quadratic function it helps to find the ___________ Recall vertex form: ______________________ where the vertex is the point ________ An inverse function would exist for the domain ____________ vertex f x = x−h 2 +k (h,k) x≥h

Restricted Domain for Inverse of Quadratics Given the functions below, identify the vertex and state the domain of the inverse: 𝑦= 𝑥−5 2 +7 Vertex at __________ Domain of inverse: _________ 𝑦= 𝑥+4 2 −9 Vertex at __________ Domain of inverse: ___________ 𝑦= 𝑥−3 2 +11 (5, 7) 𝑥≥5 𝑥≥−4 (−4, −9) (3, 11) 𝑥≥3

Rearranging an equation into vertex form To identify the vertex of a quadratic equation not in vertex form, ___________________ by forming a perfect square trinomial Many quadratic equations need to be in _______________ to find the inverse complete the square vertex form

Rearrange the Quadratic Equation Below so that it is in vertex form Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist: Find the number to create a perfect square trinomial. −6 2 2 = 𝑥 2 −6𝑥+21=0 9 Add and subtract this number. (𝑥 2 −6𝑥+9)−9+21=0 Re-write as perfect square and simplify. 𝑥−3 2 +12=0 Vertex at________ (3,12) Inverse for ________ x≥3

Rearrange the Quadratic Equation Below so that it is in vertex form Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist: Find the number to create a perfect square trinomial. 10 2 2 = 𝑥 2 +10𝑥−11=0 25 Add and subtract this number. (𝑥 2 +10𝑥+25)−25−11=0 Re-write as perfect square and simplify. 𝑥+5 2 −36=0 Vertex at________ (−5,−36) Inverse for ________ x≥−5

Rearrange the Quadratic Equation Below so that it is in vertex form Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist: Find the number to create a perfect square trinomial. 𝑥 2 +4𝑥+6=0 4 2 2 = 4 Add and subtract this number. (𝑥 2 +4𝑥+4)−4+6=0 Re-write as perfect square and simplify. 𝑥+2 2 +2=0 Vertex at________ (−2,2) Inverse for ________ x≥−2

Rearrange the Quadratic Equation Below so that it is in vertex form Rearrange the Quadratic Equation Below so that it is in vertex form. Then, identify a domain on which an inverse function would exist: Find the number to create a perfect square trinomial. 𝑥 2 −12𝑥−9=0 −12 2 2 = 36 Add and subtract this number. (𝑥 2 −12𝑥+36)−36−9=0 Re-write as perfect square and simplify. 𝑥−6 2 −45=0 Vertex at________ (6,−45) Inverse for ________ x≥6

Find the inverse of 𝑓 𝑥 = 𝑥 2 +4𝑥−9 𝑥 2 +4𝑥−9=0 𝑥 2 +4𝑥+4−4−9=0 𝑦+2 2 −13=𝑥 𝑥+2 2 −13=0 +13 +13 10000 10000 𝑥+2 2 =𝑥+13 𝑥+2=± 𝑥+13 −2 −2 𝑓 −1 𝑥 =−2± 𝑥+13

Find the inverse of 𝑦= 𝑥+3 2 −5 𝑥= 𝑦+3 2 −5 +5 +5 10000 10000 𝑥+5= 𝑦+3 2 ± 𝑥+5 =𝑦+3 −3 −3 𝑓 −1 𝑥 =−3± 𝑥+5