FINDING INVERSES OF LINEAR FUNCTIONS Original relation Inverse relation x –2 –1 0 1 2 x 4 2 0 –2 –4 y 4 2 0 –2 –4 y –2 –1 0 1 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 theoriginal relation. DOMAIN DOMAIN RANGE RANGE
FINDING INVERSES OF LINEAR FUNCTIONS Original relation Inverse relation x 4 2 0 – 2 – 4 0 4 2 – 2 – 2 – 1 1 2 0 – 4 x – 2 – 1 0 1 2 y – 2 – 1 0 1 2 y 4 2 0 – 2 – 4 – 2 0 4 0 – 1 1 2 2 – 2 – 4 Graph of original relation y = x Reflection in y = x Graph of inverserelation
FINDING INVERSES OF LINEAR FUNCTIONS To find the inverse of a relation that is given by anequation in x and y, switch the roles of x and y andsolve for y (if possible).
Finding an Inverse Relation x y 4 1 x+ 2 = y 2 2 1 The inverse relation isy = x+ 2. 2 Find an equation for the inverse of the relation y= 2 x – 4. SOLUTION y= 2 x – 4 Write original relation. x= 2 y– 4 Switch x andy. x+ 4 = 2y Add 4to each side. Divide each side by 2. If both the original relation and the inverse relation happen to befunctions, 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)) = xand 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 m0, the inverse is itself a linear function.
Verifying Inverse Functions Verify that f(x) = 2x – 4 and g(x) = x+ 2 are inverses. 1 1 1 1 ( ) f(g(x)) = fx+ 2 g(f(x)) = g(2x – 4) ( ) 2 2 2 2 = 2 x+ 2 – 4 = (2x – 4) + 2 = x + 4 – 4 = x – 2 + 2 = x = x SOLUTION Show that f (g(x)) = x andg(f (x)) = x.
Finding an Inverse Power Function x 0 ±x= y FINDING INVERSES OF NONLINEAR FUNCTIONS Find the inverse of the function f(x) = x2. SOLUTION f(x) = x2 Write original function. Replace original f(x)withy. y = x2 x = y2 Switch x and y. Take square roots of each side.
f(x) =x2 g(x) =x3 Notice that the graph of f(x) =x2can be intersected twice with a horizontal line and that its inverse is not a function. inverse ofg(x) =x3 inverse off(x) =x2 On the other hand, the graph ofg(x) =x3cannot be intersected twice with a horizontal line and its inverse is a function. g–1(x)= x 3 FINDING INVERSES OF NONLINEAR FUNCTIONS The graphs of the power functions f(x) =x2andg(x) =x3are shown along with their reflections in the liney=x. Notice that the inverse ofg(x) =x3 is a function, but that the inverse off(x) =x2is not a function. g(x)=x3 f(x)=x2 x=y2 If the domain off(x) = x2is restricted, say to only nonnegative numbers, then the inverse offis a function.
HORIZONTAL LINE TEST FINDING INVERSES OF NONLINEAR FUNCTIONS If no horizontal line intersects the graph of a functionf more than once, then the inverse off is itself a function.
Modeling with an Inverse Function ASTRONOMYNear 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= (9.01X 1026) t3where tis the age (in years) of the nebula.Write the inverse function that gives the age ofthe nebula as a function of its volume.
Modeling with an Inverse Function V =t3 9.01X 1026 V =t 3 9.01X 1026 (1.04X 10– 9) V=t 3 Volume V can be modeledby V = (9.01 X1026)t3 Write the inverse function that gives the age of the nebula as a function of its volume. SOLUTION V= (9.01X 1026)t3 Write original function. Isolate power. Take cube root of each side. Simplify.
Modeling with an Inverse Function 3 t= (1.04X 10–9) V 3 = (1.04X 10–9) 1.5X 1038 Determine the approximate age of the Ring Nebula given that its volume is about 1.5 X1038 cubic kilometers. SOLUTION To find the age of the nebula, substitute 1.5 X 1038for V. Write inverse function. Substitute for V. 5500 Use calculator. The Ring Nebula is about 5500 years old.
Homeworkp426 (16-18,27-29,33-35,42-47)PROFICIENCY REVIEW: Data Analysis (9-12)