Original relationInverse relation y420– 2– 2– 4– 4 x210– 1– 1– 2– 2 RANGE F INDING I NVERSES OF L INEAR F UNCTIONS x420– 2– 2– 4– 4 y210– 1– 1– 2– 2 An inverse relation maps the output values back to their original input values. This means that the domain of the inverse relation is the range of the original relation and that the range of the inverse relation is the domain of the original relation. RANGE DOMAIN

F INDING I NVERSES OF L INEAR F UNCTIONS x y420– 2– 4 210– 1– 2 Original relation x420– 2– 4 y210– 1– 2 Inverse relation Graph of original relation Reflection in y = x Graph of inverse relation y = x – – – – 4 2

F INDING I NVERSES OF L INEAR F UNCTIONS To find the inverse of a relation that is given by an equation in x and y, switch the roles of x and y and solve for y (if possible).

Finding an Inverse Relation Find an equation for the inverse of the relation y = 2 x – 4. y = 2 x – 4 Write original relation. S OLUTION Divide each side by 2. 2 x + 2 = y 1 2 The inverse relation is y = x If both the original relation and the inverse relation happen to be functions, the two functions are called inverse functions. Switch x and y. x y Add 4 to each side. 4 x + 4 = 2 y x = 2 y – 4

Finding an Inverse Relation I N V E R S E F U N C T I O N S Functions f and g are inverses of each other provided: f ( g ( x )) = x and g ( f ( x )) = x The function g is denoted by f – 1, read as “ f inverse. ” Given any function, you can always find its inverse relation by switching x and y. For a linear function f (x ) = mx + b where m  0, the inverse is itself a linear function.

Verifying Inverse Functions S OLUTION Show that f ( g (x)) = x and g (f (x)) = x. g ( f (x)) = g (2x – 4) = (2x – 4) + 2 = x – = x 1 2 Verify that f (x) = 2 x – 4 and g (x) = x + 2 are inverses. 1 2 f ( g (x)) = f x () = 2 x + 2 – 4 = x + 4 – 4 = x 1 2 ()

Take square roots of each side. F INDING I NVERSES OF N ONLINEAR F UNCTIONS Finding an Inverse Power Function Find the inverse of the function f (x) = x 2. S OLUTION f (x) = x 2 y = x 2 x = y 2 ± x = y Write original function. Replace original f (x) with y. Switch x and y. x  0

Notice that the inverse of g (x) = x 3 is a function, but that the inverse of f ( x ) = x 2 is not a function. If the domain of f ( x ) = x 2 is restricted, say to only nonnegative numbers, then the inverse of f is a function. F INDING I NVERSES OF N ONLINEAR F UNCTIONS The graphs of the power functions f ( x ) = x 2 and g (x) = x 3 are shown along with their reflections in the line y = x. g (x) = x 3 g (x ) = x 3g (x ) = x 3 f ( x ) = x 2 f (x ) = x 2f (x ) = x 2 inverse of g (x) = x 3 g –1 ( x ) = x 3 inverse of f ( x ) = x 2 x = y 2x = y 2 On the other hand, the graph of g (x) = x 3 cannot be intersected twice with a horizontal line and its inverse is a function. Notice that the graph of f ( x ) = x 2 can be intersected twice with a horizontal line and that its inverse is not a function.

F INDING I NVERSES OF N ONLINEAR F UNCTIONS H O R I Z O N T A L L I N E T E S T If no horizontal line intersects the graph of a function f more than once, then the inverse of f is itself a function.

Modeling with an Inverse Function A STRONOMY Near the end of a star’s life the star will eject gas, forming a planetary nebula. The Ring Nebula is an example of a planetary nebula. The volume V (in cubic kilometers) of this nebula can be modeled by V = (9.01 X ) t 3 where t is the age (in years) of the nebula. Write the inverse function that gives the age of the nebula as a function of its volume.

Modeling with an Inverse Function S OLUTION Write original function. Isolate power. Take cube root of each side. V = (9.01 X ) t 3 V 9.01 X = t 3= t 3 Simplify. (1.04 X 10 – 9 ) V = t 3 Volume V can be modeled by V = (9.01 X ) t 3 Write the inverse function that gives the age of the nebula as a function of its volume. V 9.01 X = t= t

Determine the approximate age of the Ring Nebula given that its volume is about 1.5 X cubic kilometers. To find the age of the nebula, substitute 1.5 X for V. Write inverse function. Substitute for V.  5500 Use calculator. The Ring Nebula is about 5500 years old. Modeling with an Inverse Function S OLUTION = (1.04 X 10 – 9 ) 1.5 X t = (1.04 X 10 – 9 ) V 3