Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 1 Herrega kutaa 11: Hariiroo fi Faankishinii

Wanti fuula kana irra jiru hundi boqonnaa 1 kitaaba Herrega kutaa 11 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

8
Kaardii: 10
Gaaffii shaakalaa boqonnaa 1: 1–82
Fuula kitaabaa boqonnaa 1

Boqonnaan 1 maal haguuga

1.Relations
2.Inverse of Relations and Their Graphs
3.Types of Functions
4.Composition of Functions
5.Inverse Functions and their Graphs
6.Applications of Relations and Functions

Yaadannoo gabaabaa boqonnaa 1

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

Faankishinii Gatii QulqulluuFoormulaa

|a| = \begin{cases} a & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -a & \text{if } a < 0 \end{cases}

|a| = \begin{cases} a & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -a & \text{if } a < 0 \end{cases}

$a$: lakkoofsa riilii $dimensionless$
$|a|$: gatii qulqulluu a $dimensionless$

Yoom akka itti fayyadamtu

Gatii qulqulluu lakkoofsa riilii kamiyyuu shallaguuf yookiin himoota mallattoo keessa jiran salphisuuf itti fayyadami.

Of eeggadhu

Gatiin qulqulluu lakkoofsa riilii kamiyyuu yoomiyyuu negaatiivii miti.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Domain fi Range HariirooSeera

Yoom akka itti fayyadamtu

Hariiroo akka tuuta cimdii qindaa'aatti kenname irraa domain fi range adda baasuuf kanatti fayyadamaa.

Of eeggadhu

Domain qabxiilee duraa qofa kan qabatu yoo ta'u range immoo qabxiilee lammaffaa qofa qabata. Elementoota irra deddeebi'an hin barreessinaa.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Walqixxummaa Cimdii Qindaa'aaDogoggora baratame

Yoom akka itti fayyadamtu

Cimdoonni qindaa'oon lama walqixa ta'uu isaanii mirkaneessuuf yookiin qabxiilee hin beekamne baruuf kanatti fayyadamaa.

Of eeggadhu

Cimdii qindaa'aa (x, y) keessatti walduraa duubni murteessaadha. (x, y) fi (y, x) kan walqixxaatan yoo x fi y walqixa ta'an qofaadha.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Giraafii Faallaa FaankishiniiMala

Yoom akka itti fayyadamtu

Giraafii faankishinii kenname irraa giraafii faallaa faankishinii fakkessuuf itti fayyadami.

Of eeggadhu

Qabxiilee x fi y wal jijjiiruudhaan sarara y = x irratti calaqqisiisi.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Ulaagaa Faallaa FaankishiniiSeera

Yoom akka itti fayyadamtu

Faankishinoonni kennaman f fi g faallaa walii ta'uu isaanii mirkaneessuuf itti fayyadami.

Of eeggadhu

Qindoomina lamaanuu mirkaneessuu qabda. Kallattii tokko qofa agarsiisuun ga'aa miti.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Dogoggora Mallattoo Faallaa FaankishiniiDogoggora baratame

Yoom akka itti fayyadamtu

Hiika mallattoo faallaa faankishinii f^{-1}(x) hubachuuf itti fayyadami.

Of eeggadhu

Faallaan faankishinii f^{-1}(x) -f(x) yookiin 1/f(x) wajjin wal qixa miti.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Faankishinoota Paawarii Raashinaalii NegaatiiviiGabatee wabiif(x) = x^-1/n · f(x) = x^-m/n

Faankishinii	m	n	Dimeenii	Reenjii
$f(x) = x^{-1/n}$	$1$	even	$(0, \infty)$	$(0, \infty)$
$f(x) = x^{-1/n}$	$1$	odd	$\mathbb{R} \setminus \{0\}$	$\mathbb{R} \setminus \{0\}$
$f(x) = x^{-m/n}$	odd	even	$(0, \infty)$	$(0, \infty)$
$f(x) = x^{-m/n}$	even	odd	$\mathbb{R} \setminus \{0\}$	$(0, \infty)$

Yoom akka itti fayyadamtu

Gabatee kanaan dimeenii fi reenjii faankishinoota paawarii raashinaalii negaatiivii beeki.

Of eeggadhu

Zeeroon yoomiyyuu dimeenii paawarii negaatiivii keessa hin jiru sababni isaas zeeroon qooduun hin danda'amu.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Faankishinoota Paawarii Raashinaalii PoosatiiviiGabatee wabiif(x) = x^1/n · f(x) = x^m/n

Faankishinii	m	n	Dimeenii	Reenjii
$f(x) = x^{1/n}$	$1$	even	$[0, \infty)$	$[0, \infty)$
$f(x) = x^{1/n}$	$1$	odd	$\mathbb{R}$	$\mathbb{R}$
$f(x) = x^{m/n}$	odd	even	$[0, \infty)$	$[0, \infty)$
$f(x) = x^{m/n}$	even	odd	$\mathbb{R}$	$[0, \infty)$

Yoom akka itti fayyadamtu

Gabatee kanatti fayyadamuun dimeenii fi reenjii faankishinoota paawarii raashinaalii poosatiivii argadhu.

Of eeggadhu

Giddugaleessi n lakkoofsa kophxoo yoo ta'e, lakkoofsi negaatiiviin dimeenii keessa hin galu.

Grade 11 Mathematics, fuula 1-82 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 1

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

Gaaffii 1

Find the domain of the relation R = {(45, 65), (48, 68), (41, 62), (46, 66), (50, 70)}.

A.{41, 45, 46, 48, 50}
B.{62, 65, 66, 68, 70}
C.{41, 45, 46, 48, 50, 62, 65, 66, 68, 70}
D.{45, 48, 41, 46, 50, 65}

Deebii fi sababa agarsiisi

Deebii: A {41, 45, 46, 48, 50}

Sababa

The domain is the set of all first components of the ordered pairs. The first components are 41, 45, 46, 48 and 50. The set {62, 65, 66, 68, 70} is the range, not the domain.

Gaaffii 2

Let R = {(1, 3), (2, 5), (3, 7), (4, 11), (5, 23)}. Which set is the inverse relation R^-1?

A.{(3, 1), (5, 2), (7, 3), (11, 4), (23, 5)}
B.{(1, 3), (2, 5), (3, 7), (4, 11), (5, 23)}
C.{(1, 2), (3, 4), (5, 6), (7, 8), (9, 10)}
D.{(-1, -3), (-2, -5), (-3, -7), (-4, -11), (-5, -23)}

Deebii fi sababa agarsiisi

Deebii: A {(3, 1), (5, 2), (7, 3), (11, 4), (23, 5)}

Sababa

To form the inverse, swap the order of the two components in each ordered pair. For example, (1, 3) becomes (3, 1). Doing this for all five pairs gives option A.

Gaaffii 3

Which of the following is not a power function?

A.f(x) = 3x2
B.g(x) = 5x3
C.n(x) = -4x-3
D.h(x) = 2x

Deebii fi sababa agarsiisi

Deebii: D h(x) = 2x

Sababa

A power function has the form f(x) = ax^r, where a is a nonzero real number and r is a real number. In h(x) = 2^x, the variable is in the exponent, so it is an exponential function, not a power function. The other three have the form a times x to a power.

Gaaffii 4

Using the vertical line test, a vertical line cuts the graph of a relation at two different points. What does this tell you?

A.The relation is a function.
B.The relation is not a function.
C.The relation has no domain.
D.The relation is its own inverse.

Deebii fi sababa agarsiisi

Deebii: B The relation is not a function.

Sababa

A set of points represents a function if and only if no two points are on the same vertical line. If a vertical line touches the graph twice, one first coordinate has two second coordinates, so the relation is not a function.

Gaaffii 5

Abebe buys teff at a shop in Addis Ababa. The relation R = {(2, 80), (3, 120), (2, 95), (4, 160)} lists pairs of (kilograms of teff, birr paid). Is R a function?

A.Yes, because every first component appears only once.
B.Yes, because the second components are all different.
C.No, because 2 is paired with both 80 and 95.
D.No, because 4 is paired with 160.

Deebii fi sababa agarsiisi

Deebii: C No, because 2 is paired with both 80 and 95.

Sababa

A relation is a function when each first component is related to exactly one second component. Here the first component 2 is paired with two different values, 80 and 95. So R is not a function. The pair (4, 160) does not cause the problem.

Gaaffii 6

Let R = {(x, y) : y = x² and x is in {-2, -1, 0, 1, 2}}. Find the range of R.

A.{-2, -1, 0, 1, 2}
B.{0, 1, 4}
C.{1, 4}
D.{-4, -1, 0, 1, 4}

Deebii fi sababa agarsiisi

Deebii: B {0, 1, 4}

Sababa

Substitute each x value. (-2, 4), (-1, 1), (0, 0), (1, 1) and (2, 4) are the elements of R. The second coordinates are 4, 1, 0, 1 and 4. Written as a set without repeats, the range is {0, 1, 4}.

Gaaffii 7

What is the range of the power function f(x) = x⁴?

A.R, all real numbers
B.(0, ∞)
C.[0, ∞)
D.[0, 4]

Deebii fi sababa agarsiisi

Deebii: C [0, ∞)

Sababa

For an even natural number n, the function xⁿ has domain R and range [0, ∞). Since 4 is even, f(x) = x⁴ has range [0, ∞). The range includes 0, because x = 0 gives 0.

Gaaffii 8

A square is cut out of a cardboard. Each side has length x. Which function gives the area of the square in terms of x?

A.A(x) = 4x
B.A(x) = x2
C.A(x) = x3
D.A(x) = 2x2

Deebii fi sababa agarsiisi

Deebii: B A(x) = x2

Sababa

The area of a square is side times side, so the area is x times x, which is x². This is a power function with a = 1 and r = 2. The value 4x would be the perimeter, not the area.

Gaaffii 9

Let R = {(x, y) : x, y ∈ R and y = 3x - 5}. Find the inverse relation R^-1, written as a formula for y in terms of x.

A.y = (x + 5)/3
B.y = 3x + 5
C.y = (x - 5)/3
D.y = x/3 - 5

Deebii fi sababa agarsiisi

Deebii: A y = (x + 5)/3

Sababa

To find the inverse, swap x and y to get x = 3y - 5. Then solve for y: add 5 to both sides to get x + 5 = 3y, and divide by 3 to get y = (x + 5)/3. Option C is wrong because the 5 must be added, not subtracted.

Gaaffii 10

Let R = {(-3, 2), (-2, 3), (-1, 4), (0, 5), (1, 6), (2, 7), (3, 8)}. Find the domain and the range of R^-1.

A.Domain {2, 3, 4, 5, 6, 7, 8}; range {-3, -2, -1, 0, 1, 2, 3}
B.Domain {-3, -2, -1, 0, 1, 2, 3}; range {2, 3, 4, 5, 6, 7, 8}
C.Domain {-3, -2, -1, 0, 1, 2, 3}; range {-3, -2, -1, 0, 1, 2, 3}
D.Domain {2, 3, 4, 5, 6, 7, 8}; range {2, 3, 4, 5, 6, 7, 8}

Deebii fi sababa agarsiisi

Deebii: A Domain {2, 3, 4, 5, 6, 7, 8}; range {-3, -2, -1, 0, 1, 2, 3}

Sababa

The domain of the inverse is the range of R, and the range of the inverse is the domain of R. The range of R is {2, 3, 4, 5, 6, 7, 8}, and the domain of R is {-3, -2, -1, 0, 1, 2, 3}. So the domain of R^-1 is {2, 3, 4, 5, 6, 7, 8} and its range is {-3, -2, -1, 0, 1, 2, 3}.

Appiin boqonnaa kana irratti gaaffii 12 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.