Math-3Lesson 3-4

The Reciprocal Function

VocabularyDomain: The set of input values that the relation

“maps” to output values in the real number system.

Domain

xy 1

What is the domain?

or

What values of ‘x’ will result in the

output value being a real number ?

Domain xy 1

‘radicand’ ≥ 01 - x ≥ 0

Solving for x: + x

1 ≥

+ x

x

1x ≤

x : x ≤ 1Domain of isxy 1

“The domain is all values of ‘x’ such that x ≤ 1”

Domain

15 xy‘radicand’ ≥ 0

5x - 1 ≥ 0Solving for x: + 1

5x ≥

+ 1

1

1/5x ≥

x : x ≥ 1/5Domain of is15 xy

Domain: The input values that the relation

“maps” to output values in the real number system

3

2

x

xy

2 xy Domain: all real #’s

Denominator ≠ 0

Domain: x ≠ 3

Domain: (-∞,3) U (3,∞)

Domain: all real #’s except 3

Domain

32 x

2

3x

032 x

),2

3[

‘radicand’ ≥ 0

Domain

63

25

x

x

063 x

63 x

U ,22,

Denominator ≠ 0

2x

‘radicand’ ≥ 0

What is the domain?3

3

x

x

3x

03 x 03 x

3x

),3(

Denominator ≠ 0

Must satisfy both conditions

3x

‘radicand’ ≥ 0

What is the domain?1

5

x

x

5x

05 x 01x

1x

),5[

Denominator ≠ 0

Must satisfy both conditions

5x

Reducing Rational Expressions

1

122

x

xx

1 x

Factor, factor, factor (then use inverse property

of multiplication)

1

)1)(1(

x

xx

Domain of ORIGINAL

expression (NOT the same)

as the simplified version!!!

+ x

+ y

- y

- x

There is a “hole” at the

“excluded value of x”

Your Turn: Simplify the equation. What is the

domain?

4

442

2

x

xxy

)2)(2(

)2)(2(

xx

xx

)2(

)2(

x

xy

2,2 :Domain x

There is a “hole” in the graph at x = -2

The excluded values of ‘x’ whose factors “disappear” due to

simplification (using the inverse property of multiplication)

end up as holes in the graph of the simplified equation.

Your Turn: Simplify the equation. What is the

domain? Where are the holes”

33

)1(6 3

x

xy

)1(3

)1)(1(*2*3 2

x

xx

2)1(2 xy1 :Domain x

There is a hole at x = 1

because the factor (x – 1)

causes division in the

original function but is

eliminated by simplification

using the inverse property

of multiplication.

Your Turn: Simplify the equation. What is the domain?

36

361232

23

x

xxxy

)6)(6(

)124(3 2

xx

xxx

)6)(6(

)2)(6(3

xx

xxx

)6(

)2(3

x

xxy

6,6 :Domain x

There is a hole at x = -6

)66(

)26)(6(3)6(

f 12)6( f

There is a hole at (-6, -12)

Vocabulary

Asymptote: A vertical, horizontal, or oblique

line that the graph approaches but NEVER

reaches.

Asymptotes are not part of the graph but you

can see them easily.

xxf

1)(

Parent function: the simplest function in a family of

functions.

x y

1/10 = 0.1

1/5 = 0.2

1

5

10

10

5

12.051

1.010

1

0 ?

x f(x)1/10=0.1

1/5=0.2

0

1

5

10

Reciprocal Function

10

5

xxf

1)(

??

1

1/5=0.21 2 3 4 5 6 7 8 9 10

10

5

-3 -2 -1

1/10=0.1

x = 0

xxf

1)(

Your turn:

What will happen to the reciprocal function if you

add 3 to it?

y = 3

31

)( x

xfThe new horizontal

asymptote is: y = 3

What value is not

part of the domain?

What value is not

part of the range?

x = 0

y = 3

x f(x)1

Why is there a vertical asymptote?

1

10

xxf

1)(

1 2 3 4 5 6 7 8 9 10

10

5

-3 -2 -1

0.1

1000.01

0.001 1000

410 000,10104

1210 1210

?2

0 0

2

1*0

?0

2 undefined

Write a “rule” about what the fraction equals based

upon where the zero is (numerator and denominator).

Why is there a horizontal asymptote?

xxf

1)(

Can the denominator EVER make the value of the fraction

equal to zero?

What part of the fraction makes the fraction equal to zero?

The output of the function can NEVER equal zero

horizontal asymptote at y = 0

x = 0

xxf

1)(

f(x)

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

-10 -5 0 5 10

f(x)

The Reciprocal Function

y = 0

What value is not

part of the domain?

What value is not

part of the range?

x = 0

xxf

1)(

Your turn: What will happen to the reciprocal function if you subtract 2

from it?

y = -22

1)(

xxf

What is the new

horizontal asymptote?

What is the domain and range

of the new function?

x = 4

xxf

1)(

Your turn: What will happen to the reciprocal function if you replace ‘x’

with ‘x – 4’ ?

y = 0)4(

1)(

xxf

The new vertical asymptote is: x = 4

The “Reciprocal” function

xxf

1)(

)(4)( xfxg )(8)( xfxh

Parent

Function

xxg

4)(

xxh

8)(

The “Reciprocal” function

xxf

1)(

)(12)( xfxk )(2

1)( xfxk

Parent

Function

xxk

12)(

xxk

5.0)(

xxk 2

1

)(

xxk

2

1)(

How is the graph of the parent function

is transformed by each of the following equations?

71

)( x

xg

)2(

5)(

xxh

xxf

1)(

5)3(

3)(

xxf

Up 7

VSF = 5, Right 2

Reflected across the x-axis,

VSF = 3

Left 3

Up 5

What are the horizontal and vertical asymptotes for each>

71

)( x

xg

)2(

1)(

xxh

5)3(

3)(

xxf

HA: y = 8

VA: x = 0

HA: y = 0

VA: x = 2

HA: y = 5

VA: x = -3

Your turn:What are the horizontal and vertical asymptotes for each

of the following reciprocal functions?

5.01

)( x

xg

)6(

1)(

xxh

7)2(

1)(

xxf

Asymptotes are lines. I want the equation of the line.

y = 0.5

y = -7

x = 0

x = 2

y = 0x = -6