inverse functions [repaired]

INVERSE FUNCTIONS

Prepared by: RAPHAEL V. PEREZ, CpE

INVERSE FUNCTIONS

• In short, the reflector of the original function at the radical axis y = x

• The original function is

f(x) and then the inverse function of f(x) is:

f-1(x) or F(x) in other books

INVERSE FUNCTIONS

• In terms of ordered pairs, the inverse of

f(x) = (a,b) is

f-1(x) = F(x) (b,a)

• In short, the inverse of the set:

f(x) = (a1,b1), (a2,b2), (a3,b3),…, (an+1,bn+1)

is

f-1 (x) = F(x) = (b1, a1), (b2,a2), (b3,a3),…, (bn+1,an+1)

y = f(x)

(a1, b1)(a2, b2)

(a3, b3)

(an+1, bn+1)

y = f(x)

(a1, b1)(a2, b2)

(a3, b3)

(an+1, bn+1)

(b1, a1)

(b2, a2)

(b3, a3)

(bn+1, an+1)

The inverse off(x):

f-1(x) = F(x)

The set of ordered pairs

at f(x) has been inverted

INVERSE FUNCTIONS

EXAMPLE : FIND THE INVERSE FUNCTION OF THE FOLLOWING:

f(x) = (-2,-6), (2,-4), (6,-2), (10,0)

Ans:

f-1(x) = (-6,-2), (-4,2), (-2,6), (0,10)

-16 -14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14 16

-6

-4

-2

2

4

6

8

x

yAxis y = x

To reflect graph about axis y = x, 1) Go to Function | Custom Functions... and define your function F(x).2) Click Calc | Animate... | Animate.

f(x) = (-2,-6), (2,-4), (6,-2), (10,0)

f-1(x) = (-6,-2), (-4,2), (-2,6), (0,10)

(-2,-6)

(2,-4)

(6,-2)(10,0)

(-6,-2)

(-4,2)

(-2,6)

(0,10)

INVERSE FUNCTIONSNow, in terms of POLYNOMIAL FUNCTION. Here are the steps to get the inverse function [f-1(x)] of the original function f(x):

1. Change f(x) to “y” on the given function.

2. Invert the variables between x and y. The y variable in (1) will be “x” and for x variable on right side will be “y”.

3. Solve for y from (2).

4. Change “y” into f-1(x).

5. Solve for f [f-1(x)] and f-1[f(x)] (Composition Method). The answer must be “x”.

INVERSE FUNCTIONSExample 1:

INVERSE FUNCTIONSSolution:

Step 1: Change f(x) to “y” on the given function.

INVERSE FUNCTIONSStep 2: Invert x and y: y becomes “x” and x becomes “y”

becomes

INVERSE FUNCTIONSStep 3: Solve for y from number 2 step.

2 (To cancel denominator: LCD is 2)

2

INVERSE FUNCTIONSStep 4: Change y to f-1(x).

will be

So, the inverse function of is

INVERSE FUNCTIONSStep 5: Get the Composition

)(x) and )(x)

INVERSE FUNCTIONS

For )(x)

INVERSE FUNCTIONS

For )(x)

INVERSE FUNCTIONSWhen you get “x” on the composition method,

meaning our answer is correct.

GRAPH!

𝑓 (𝑥 )=𝑥− 3  2

𝑦=𝑥

𝑓 (𝑥 )=𝑥− 3  2

𝑦=𝑥

f(x)=2x+3

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-3

-2

-1

1

2

3

4

5

x

y

Axis y = x

To reflect graph about axis y = x, 1) Go to Function | Custom Functions... and define your function F(x).2) Click Calc | Animate... | Animate.

𝑓 (𝑥 )=𝑥− 3  2

𝑦=𝑥

INVERT THE ORDERED PAIRS FROM f(x) to

graph(no need to solve)

P1 P2

x 0 3

y - 0

P1 P2

x - 0

y 0 3 (3,0)

(0, -3/2)

(0,3)

(-3/2, 0)

INVERSE FUNCTIONS

Answers to be needed:

1. The Inverse Function Equation

2. The Composition: )(x) and )(x)

3. The graph

Note: You should know the topic the transformation of variables by knowing the properties of algebra.

INVERSE FUNCTIONSExample 2:

solution:

Step 1:

INVERSE FUNCTIONSExample 2:

solution:

Step 2:

INVERSE FUNCTIONSExample 2:

Step 3: Solve for y:

INVERSE FUNCTIONSExample 2:

Step 4: Change “y” to f--1(x):

INVERSE FUNCTIONS

Step 5: Composition :

)(x) and )(x)

INVERSE FUNCTIONS

Step 5: Composition :

For )(x)

(x)] = x + 4 – 4

= x

INVERSE FUNCTIONS

Step 5: Composition :

For )(x)

[f(x)]

=

= x

INVERSE FUNCTIONS

POSSIBLE TO GRAPH ?

You may use the graphical software for Cartesian and Polar coordinates

CLICK HERE

INVERSE FUNCTIONSQUESTIONS?

For graphing: you will graph only linear functions ( y = mx + b).

Other functions like:

exponential (y = bx)

logarithmic (y = logb x or y = ln x) ,

trigonometric (y = a sin x)

and second degree or higher polynomials

(y = axn + xn-1 +…+a0)

are not yet discussed for way of sketching the function, sometimes you need to use programmable and graphical calculators or the

computers. It’s hard to sketch the mentioned functions.

INVERSE FUNCTIONS

If you want this application program for graphing purposes

install on your Personal Computer,

visit www.padowan.dk

this is a free-download software program.

INVERSE FUNCTIONSExercises: Find the Inverse function (x) of each function and verify it by

computing the composition for )(x) and )(x). Graph it.

*Graphical software is needed.

5

45 xxf

25

4

51 x

xf

COMBINATION OF OPERATIONS OF FUNCTIONS

Prepared by: RAPHAEL V. PEREZ

RECALL: OPERATION OF FUNCTIONS

ADDITION/SUBTRACTION:

MULTIPLICATION:

DIVISION: ;

COMPOSITION: ○

○

Note: ○○

EVALUATE THE FUNCTIONSEXAMPLE 1:

EVALUATE:

EVALUATE THE FUNCTIONSSOLUTION:Given functions:

Ans.

EVALUATE THE FUNCTIONSSOLUTION:

EVALUATE:

Ans.

EVALUATE THE FUNCTIONSSOLUTION:

EVALUATE:

Ans

EVALUATE THE FUNCTIONSSOLUTION:

EVALUATE:

Recall the answers:

So:

EVALUATE THE FUNCTIONSSOLUTION:

EVALUATE:

Recall the answers:

So:

EVALUATE THE FUNCTIONS

So: