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Inverse Relations and Functions OBJ: Find the inverse of a relation Draw the graph of a function and its inverse Determine whether the inverse of a function is a function FINDING INVERSES OF LINEAR FUNCTIONS 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. Original relation x y – 2 DOMAIN –1 0 1 4 Inverse relation 2 x RANGE 2 0 –2 –4 y 4 DOMAIN 2 0 –2 –4 – 2 RANGE –1 0 1 2 FINDING INVERSES OF LINEAR FUNCTIONS Original relation x –2 –1 0 y 4 2 1 Inverse relation 2 0 –2 –4 Graph of original relation Reflection in y = x Graph of inverse relation x 4 2 0 –2 –4 y –2 –1 0 1 2 y=x FINDING INVERSES OF LINEAR FUNCTIONS 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. SOLUTION y=2x–4 x =2y – 4 x + 4 = 2y 1x+2=y 2 Write original relation. Switch x x and yy. Add 4 4 to each side. Divide each side by 2. 2 The inverse relation is y = 1 x + 2. 2 If both the original relation and the inverse relation happen to be functions, the two functions are called inverse functions. Finding an Inverse Relation INVERSE FUNCTIONS Functions f and g are inverses of each other provided: f (g (x)) = x g ( f (x)) = x and 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 Verify that f (x) = 2x – 4 and g (x) = 1 x + 2 are inverses. 2 SOLUTION Show that f (g (x)) = x and g (f (x)) = x. 1 f (g (x)) = f 2 x + 2 1 = 2 x +2 –4 2 = x+4 – 4 ( ( =x ) ) g (f (x)) = g (2x – 4) = 1 (2x – 4) + 2 2 = x–2+2 =x FINDING INVERSES OF NONLINEAR FUNCTIONS Finding an Inverse Power Function Find the inverse of the function f (x) = x 2. SOLUTION f (x) = x 2 Write original function. y = x2 Replace original x = y2 Switch x and y. ± x=y f (x) with y. Take square roots of each side. x0 FINDING INVERSES OF NONLINEAR FUNCTIONS 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. 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. On the other hand, the graph of g (x) = xf3(xcannot ) = x 2 be intersected twice with a horizontal line and its inverse is a function. gof ( xf)(x = x=3 x 2 Notice that the graph ) 3 –1 g ( x ) = x can be intersected twice with a horizontal line and that its inverse is not a function. x= y2 If the domain of f (x) = x 2 is restricted, say to only nonnegative numbers, then the inverse of f is a function. FINDING INVERSES OF NONLINEAR FUNCTIONS H O R I Z O N T A L L I N E T E ST 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 ASTRONOMY 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 10 26 ) 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 Volume V can be modeled by V = (9.01 X 10 26 ) t 3 Write the inverse function that gives the age of the nebula as a function of its volume. SOLUTION V = (9.01 X 1026 ) t 3 V 9.01 X 10 3 26 V 9.01 X 10 26 = t3 =t (1.04 X 10– 9 ) V = t 3 Write original function. Isolate power. Take cube root of each side. Simplify. Modeling with an Inverse Function Determine the approximate age of the Ring Nebula given that its volume is about 1.5 X 10 38 cubic kilometers. SOLUTION To find the age of the nebula, substitute 1.5 X 10 38 for V. t = (1.04 X 10–9 ) 3 V Write inverse function. = (1.04 X 10–9 ) 3 1.5 X 1038 Substitute for V. 5500 Use calculator. The Ring Nebula is about 5500 years old.