Download - 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: