Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 2 Herrega kutaa 11: Ibsitoota Raashinaalii fi Faankishinoota Raashinaalii

Wanti fuula kana irra jiru hundi boqonnaa 2 kitaaba Herrega kutaa 11 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

4
Kaardii: 10
Gaaffii shaakalaa boqonnaa 2: 83–130
Fuula kitaabaa boqonnaa 2

Boqonnaan 2 maal haguuga

1.Rational Expressions
2.Rational Equations and Rational Inequalities
3.Rational Functions and Their Graphs
4.Applications

Yaadannoo gabaabaa boqonnaa 2

Kaardiiwwan fuula Herrega kutaa 11 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Asimptootiiwwan Faankishinii RaashinaaliiGabatee wabiideg(p) < deg(q) · deg(p) = deg(q) · deg(p) = deg(q) + 1

Haala digrii	Gosa asimptootii	Hiriira
$\text{deg}(p) < \text{deg}(q)$	Horizontal asymptote	$y = 0$
$\text{deg}(p) = \text{deg}(q)$	Horizontal asymptote	$y = \frac{a_n}{b_n}$
$\text{deg}(p) = \text{deg}(q) + 1$	Oblique asymptote	$y = mx + b$ via polynomial division

Yoom akka itti fayyadamtu

Giraafii faankishinii raashinaalii yeroo kaasamu asimptootii dalgee yookiin kophxaa baruuf fayyadamaa.

Of eeggadhu

Asimptootiin kophxaa kan uumamu digriin lakkooftuu isa jalaa irra tokkoon yoo caale qofaadha.

Grade 11 Mathematics, fuula 83-130 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Hojiiwwan Himoota RaashinaaliiGabatee wabiiAddition · Subtraction · Multiplication · +1

Hojii	Seera	Ulaagaa
Addition	$\frac{p(x)}{q(x)} + \frac{r(x)}{s(x)} = \frac{p(x)s(x) + r(x)q(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Subtraction	$\frac{p(x)}{q(x)} - \frac{r(x)}{s(x)} = \frac{p(x)s(x) - r(x)q(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Multiplication	$\frac{p(x)}{q(x)} \cdot \frac{r(x)}{s(x)} = \frac{p(x)r(x)}{q(x)s(x)}$	$q(x)$ and $s(x)$ are not zero
Division	$\frac{p(x)}{q(x)} \div \frac{r(x)}{s(x)} = \frac{p(x)s(x)}{q(x)r(x)}$	$q(x)$ , $s(x)$ , and $r(x)$ are not zero

Yoom akka itti fayyadamtu

Ida'ama, hir'isa, baay'ata fi hirmaata himoota raashinaalii shallaguuf itti fayyadamaa.

Of eeggadhu

Hirmaata keessatti, lakkooftuun isa hirus gara jalaatti waan jijjiiramuuf zeeroo ta'uu hin danda'u.

Grade 11 Mathematics, fuula 83-130 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Hiika Himannoo SaamudaaSeera

Yoom akka itti fayyadamtu

Himannoon aljebiraa tokko himannoo saamudaa ta'uu isaa adda baasuuf itti fayyadami.

Of eeggadhu

Lakkooftuun fi gadi-qoodduun lamaanuu poolinoomiyaalii ta'uu qabu, gadi-qoodduun zeeroo ta'uu hin danda'u.

Grade 11 Mathematics, fuula 83-130 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Foormulaa HojiiFoormulaa

W = r t

W = r t

$W$: Hojii raawwatame $1$
$r$: Reetii hojii waloo $\text{h}^{-1}$
$t$: Yeroo fudhate $\text{h}$

Yoom akka itti fayyadamtu

Yeroo hojjattoonni yookiin ujummoowwan hedduun waliin ta'anii hojii tokko hojjetanitti fayyadamaa.

Of eeggadhu

Reetii hojii yeroo tokko keessatti hojjetamu ida'aa malee yeroo guutuu kallattiin hin ida'inaa.

Grade 11 Mathematics, fuula 83-130 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 2

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Across which line is the graph of a function reflected to obtain the graph of its inverse function?

A.The line y = x
B.The line y = -x
C.The line y = 0 (x-axis)
D.The line x = 0 (y-axis)

Deebii fi sababa agarsiisi

Deebii: A The line y = x

Sababa

The graph of the inverse of a given function is obtained by reflecting the graph of the function with respect to the line y = x.

Gaaffii 2

By definition, an expression written in the form p(x) / q(x) is a rational expression if p(x) and q(x) are what type of expressions, with q(x) ≠ 0?

A.Polynomial expressions
B.Absolute value expressions
C.Radical expressions
D.Logarithmic expressions

Deebii fi sababa agarsiisi

Deebii: A Polynomial expressions

Sababa

A rational expression is defined as a quotient of two polynomials, p(x) / q(x), where q(x) ≠ 0.

Gaaffii 3

If R is a relation, which of the following is always true about its inverse relation R^-1?

A.Domain of R-1 = Range of R
B.Domain of R-1 = Domain of R
C.Range of R-1 = Range of R
D.Domain of R-1 = Range of R-1

Deebii fi sababa agarsiisi

Deebii: A Domain of R-1 = Range of R

Sababa

For any relation R, the domain of R^-1 equals the range of R, and the range of R^-1 equals the domain of R.

Gaaffii 4

What is the value of the signum function sgn(x) when x > 0?

A.1
B.0
C.-1
D.x

Deebii fi sababa agarsiisi

Deebii: A 1

Sababa

By definition, sgn(x) = 1 if x > 0, 0 if x = 0, and -1 if x < 0.

Gaaffii 5

Given the linear function f(x) = 3x - 1, what is its inverse function f^-1(x)?

A.(1/3)x + (1/3)
B.(1/3)x - (1/3)
C.-3x + 1
D.3x + 1

Deebii fi sababa agarsiisi

Deebii: A (1/3)x + (1/3)

Sababa

Let y = 3x - 1. Interchanging variables gives x = 3y - 1. Solving for y gives 3y = x + 1, so y = (1/3)x + (1/3).

Gaaffii 6

A car rental company in Addis Ababa charges a flat fee of 300 Birr plus 15 Birr per kilometer to rent a van. What is the total cost for an 80 kilometer trip?

A.1500 Birr
B.1200 Birr
C.1800 Birr
D.1450 Birr

Deebii fi sababa agarsiisi

Deebii: A 1500 Birr

Sababa

The cost function is y = 15x + 300. For x = 80, y = (15 × 80) + 300 = 1200 + 300 = 1500 Birr.

Gaaffii 7

The force applied to a spring varies directly with the distance it stretches. If applying 30 N of force stretches the spring 3 m, how far will it stretch when 5 N of force is applied?

A.0.5 m
B.1.5 m
C.2.0 m
D.0.2 m

Deebii fi sababa agarsiisi

Deebii: A 0.5 m

Sababa

Using F = kd, we find 30 = k(3), so k = 10 N/m. When F = 5 N, d = F / k = 5 / 10 = 0.5 m.

Gaaffii 8

Given the function f(x) = 2 - 3 sgn(x), what is the value of f(5)?

A.-1
B.2
C.5
D.-3

Deebii fi sababa agarsiisi

Deebii: A -1

Sababa

Since 5 > 0, sgn(5) = 1. Therefore, f(5) = 2 - 3(1) = 2 - 3 = -1.

Gaaffii 9

A vendor buys 300 newspapers at 6 Birr each. He sells 200 newspapers at 10 Birr each, and sells 50 unsold newspapers to a wastepaper dealer at 4 Birr each. What is his total profit for that day?

A.400 Birr
B.200 Birr
C.600 Birr
D.800 Birr

Deebii fi sababa agarsiisi

Deebii: A 400 Birr

Sababa

Total cost = 300 × 6 = 1800 Birr. Total revenue = (200 × 10) + (50 × 4) = 2000 + 200 = 2200 Birr. Profit = 2200 - 1800 = 400 Birr.

Gaaffii 10

In a closed economy model where E = C + I + G and equilibrium occurs at E = Y, consumption is C = 15 + 0.90Y, investment is I = 20 + 0.05Y, and government spending is G = 25. What is the equilibrium income Y?

A.1200
B.700
C.1095
D.60

Deebii fi sababa agarsiisi

Deebii: A 1200

Sababa

At equilibrium, Y = (15 + 0.90Y) + (20 + 0.05Y) + 25 = 60 + 0.95Y. Thus, Y(1 - 0.95) = 60, which gives 0.05Y = 60, so Y = 60 / 0.05 = 1200.

Appiin boqonnaa kana irratti gaaffii 18 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.

Fuula kitaabaa boqonnaa 2

Boqonnaan 2 Maxxansa Ministeera Barnootaa keessatti fuula 83 hanga fuula 130 ti.

Maxxansa Ministeera Barnootaa, Ingiliffaa, Fuula 494

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Yaadannoo gabaabaa Herrega kutaa 11 hunda

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