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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