Pc Linear Inverses

In ve rse of a Lin ear Fu nctio n

MATH PRO JEC T

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

RANGE 2 0

Inverse relation

2

x

–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

y –2 –1 0

–2 –4 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 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

The function g is denoted by  f

 g ( f –1

(x)) = x

, 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) = 2 x – 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 2x + 2 – 4 = 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 y = x2

Replace original  f (x) with y.

x = y2

Switch x and y.

±    x = y  

 

Write original function.

Take square roots of each side.

x ≥ 0

FINDING INVERSES OF NONLINEAR FUNCTIONS g (x) =

2 3 The graphs of the power functions f ((xx)) == xx 2 and g (x) 3 = x are x  

shown along with their reflections in the line y = x. Notice that the inverse of g  (x) = x 33 is a function, but that inverse of of ff ((x x)) == xx 22 is not a function. the inverse On the other hand, the graph of 2 be intersected g (x) = xf3(xcannot  ) = x  twice with a horizontal line and its inverse is a function.

gof  (xf  )(   = )x= 3 x 2 Notice that the graph x –1 3x g   (x ) =    can be intersected twice with a horizontal line and that its inverse is not a function.

x = y 2

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 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

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

Write original function.

= t3

Isolate power.

=t

Take  cube root  of each side.

(1.04 X 10– 9 ) 3 V = t

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.