Summary Inverse Functions And Picard Method (Notes & Solved Examples)

Inverse Functions and the Picard Method

As you know, a function is a rule that assigns a number in its range to each
number in its domain. Some functions, like.

y sinx. y x*, and y = 3

can give the same output for different inputs. But other functions, like

y = Vx. y = x,and y = 4x - 4.


tinvee inverse- always give different outputs for different inputs. Functions that always
give different outputs for diferent inputs are called one-to-one functions.
2.39 T he inverse of a Since each output of a one-to-one function comes from
function f sends just one input,
every output of f back to the input from any one-to-one function can be reversed to turn the outputs back into the
which it came.
inputs from which they (Fig. 2.39).
came
The function defined by reversing a one-to-one function f is called the
inverse of f. The symbol for the inverse of
f is f-, read "f inverse." The
symbol 1 in f' is not an exponent: f-"(x) does not mean 1/f(x).
-




can see in Fig. 2.40. the result of
As you
composing f and f-1 in either order is the
identity function. the function that assigns each number to itself.
Figure 2.41 shows the graph of the functign y Vx, whose domain is
=

x 0 and whose range is y 20. For each
2.40 I1 y = ((x) is a one-to-one tunction, input x, the function gives a
then f(I(x))
I(x)) == xx and
and f(1(Y)) = y. Each of single output, y VXo. Since every nonnegative is the
=
then f()
the composites f'.f and fof' is the identity xunder this function, we can reverse y image of just one
the construction. That is, we can
start with a y 2 0 and then
function on its domain. go over to the curve and down to x
X-axis. The construction in
reverse defines the
y2 on the =


inverse of f(x) function g(y) y, the =
=
Vx.
The algebraic description of what we see in Fig. 2.41 is
Yo
s(f(x)) = (Vx° = x.
f(e(y)) =
Vy =
y.
(1)
(In general, Vy =
Iyl. but since y20 in this
In inverse notation, case, we get Vy* =
y.)

8 f-1
Note that the
f(x) is
equations in (1) also say thatf is the
inverse of y = vx =
inverse of g:
2.41 The

f = 9(y). f= 8
=

, Every one-to-one function is the inverse of its own inverse:
-) = f.
y g(x) = x?
If the squaring function g is expressed with x as the independent
y
variable, so that g(x) = x*, the graph of g can be obtained by reflecting the
va graph of f in the 45° line y = x, as in Fig. 2.42. This is because any two
y fx) = v*
points(x, y) and (y, x) whose coordinates are reversed are symmetric with
respect to the line y = X. The pairs (a, Va) and ( Va, a) have this symme
try. The first lies on the graph of f(x) = Vx, the second on the graph of
8(x) = x2
If a one-to-one function is defined by an equation y = f(x) that we can
solve for x in terms of y, we get an equation in the form x = gly) that
expresses the inverse of fin term[ of y. To express the inverse in a form in
which x denotes the independent variable, we interchange the letters x
2.42 The graph of a function and its and yin the equation x = g(y). Here are some examples.
inverse are symmetric with respect to the
line y = X.
EXAMPLE 1 Find the inverse of


y =x+3.
Solution We solve the given equation for x in terms of y:
x = 4y - 12.


We then interchange x and y in the formula x = 4y - 12 to get

y = 4x - 12.


The inverse of y = }x +3 is y = 4x - 12.
To check, we let

fx)
)=X+3, g(x) = 4x - 12.




Then,
e(fx)) = 4(Gx +3-12 = x + 12 - 12 = x.

f(g(x)) = ( 4 x 12) +3 = x - 33 = x



EXAMP.E 2 Find the inverse of y = Vx.

Solution We solve the given equation for x in terms ot y to get



Then we interchange x and y to get

y x
This gives an expression for the inverse n which the independent v a r -
iable is denoted by x. To check that the functions y N and y Va are = =



in either order is the identity
inverses, we calculate that their composite
function:
Vx= x| = x and (Vx)* = x.



holds because x 0 in this example.)
(The equality Ix| = x