6-7 Inverse Relations and Functions Find the inverse of a relation or function by relating domain and range.

Inverse Relation If a relation pairs elements a in the domain with elements b in the range, its inverse pairs b with a. If (a, b) is an ordered pair in a relation, (b, a) is an ordered pair of the relation’s inverse If both a relation and its inverse are functions, they are inverse functions.

Finding the Inverse Create the table which shows the inverse of s Sketch the graph of s Now sketch the graph of the inverse Notice that the inverse is reflected across the line y = x

Finding an Equation for the Inverse What is the equation of the inverse of 𝑦= 𝑥 2 −1 Since the inverse is found by switching the domain and range, switch x and y 𝑥= 𝑦 2 −1 Now solve for y 𝑦=± 𝑥+1 You try: find the inverse of 𝑦=2𝑥+8 𝑦= 1 2 𝑥−4

Graphing Inverses The graph of the function 𝑦= 𝑥 2 −1 is a parabola with vertex (0, -1) The inverse relation is the reflection of that parabola across the line y = x.

Finding an Inverse Function Denoted 𝑓 −1 and read as “inverse of f” or “f inverse” What is 𝑓 −1 if 𝑓 𝑥 = 𝑥−2 Write as 𝑦= 𝑥−2 Now switch and solve for y: 𝑥= 𝑦−2 𝑥 2 =𝑦−2 𝑦= 𝑥 2 +2 𝑓 −1 𝑥 = 𝑥 2 +2

Domain and Range The domain of the inverse is the same as the range of the original relation. Ex: 𝑓 𝑥 = 𝑥−2 and 𝑓 −1 𝑥 = 𝑥 2 +2 The domain of 𝑓 −1 is the same as the range for f. So the domain is all real numbers x ≥0 or [0,∞) The range of 𝑓 −1 is the same as the domain for f So the range is all real numbers 𝑦≥2 or [2,∞) If each x-value maps to exactly one y-value and it is also true that each y-value maps to exactly one x-value, then the function is called a one-to-one function.

Composition of Inverse Functions If 𝑓 𝑥 = 1 𝑥−1 Find 𝑓∘ 𝑓 −1 1 Start by finding 𝑓 −1 (𝑥): 𝑓 −1 𝑥 = 1 𝑥 +1 Now evaluate 𝑓 −1 (1): 𝑓 −1 1 = 1 1 +1=2 Find 𝑓 2 = 1 2−1 =1 Now find ( 𝑓 −1 ∘𝑓)(1) Since 1 is not in the domain of f, we cannot find 𝑓(1) and 𝑓 −1 ∘𝑓 1 does not exist.

Assignment Odds p.410 #11-25, 31-35,39,41,49