3.1.2 function


1. What is the defining characteristic of a function?
A) Each element of the range has exactly one corresponding element in the domain.
B) Each element of the domain has exactly one corresponding element in the range.
C) Each element of the domain corresponds to at least two elements in the range.
D) Each element of the range has no corresponding elements in the domain.

2. In the relation R = {(5, -2), (3, 5), (3, 7)}, is R a function?
A) Yes
B) No
C) Cannot be determined
D) Depends on the domain elements

3. If a relation assigns the elements (2, 4), (3, 4), (6, -4) to the domain {2, 3, 6}, is this relation a function?
A) Yes
B) No
C) Maybe
D) Depends on the range values

4. What notation is used to denote a function from set A to set B?
A) f : A ⇾ B
B) f : A → B
C) A → B : f
D) A → f → B

5. If f(1) = 1, f(2) = 6, f(3) = 8, f(4) = 8, is this a valid function from set A = {1, 2, 3, 4} to B = {1, 6, 8, 11, 15}?
A) Yes
B) No
C) Depends on the values assigned
D) Not enough information to decide

6. What is the role of the codomain in a function?
A) It represents the set of all possible input values.
B) It represents the set of all possible output values.
C) It determines the uniqueness of the function.
D) It is the same as the range of the function.

7. If x is an element from set A and y is the corresponding image of x under function f, how is this relationship denoted?
A) x ➔ y
B) f(x) = y
C) x → f → y
D) f(x) ➔ y

8. Which of the following statements about functions is correct?
A) A function can have multiple outputs for a single input.
B) Each element in the domain can be mapped to multiple elements in the range.
C) The range of a function is always equal to the codomain.
D) The domain of a function is always a proper subset of the codomain.

9. If a function is single-valued, what does it imply?
A) Each element in the domain is mapped by the function to exactly one element in the range.
B) Multiple elements in the domain can be mapped to the same element in the range.
C) The function does not have a defined domain.
D) The function assigns random values to the elements of the range.

10. What is the set called that consists of all the image values of elements from the domain under a function?
A) Domain
B) Codomain
C) Range
D) Coimage
Answer
https://t.me/yohana1234567/1089

Real value

1. Which of the following best describes a real-valued function?
A) It maps all elements of its domain to real numbers.
B) It has a complex codomain.
C) It assigns at least one complex number in its range.
D) It operates exclusively on rational numbers.

2. Consider the functions f(x) = x² + 3x + 7 and g(x) = x. What is the result of f + g?
A) x² + 4x + 7
B) 2x² + 4x + 7
C) x² + 2x
D) x² + 3x

3. In the quotient of two functions f and g, what additional restriction must be imposed?
A) g(x) ≠ 0 for all x
B) g(x) = 0 for all x
C) f(x) ≠ 0 for all x
D) f(x) = 0 for all x

4. If f(x) = 3x² + 2 and g(x) = 5x - 4, what is the result of f * g?
A) 15x³ - 12x² + 10x - 8
B) 3x² + 5x + 18
C) 15x² - 12x + 10x - 8
D) 15x³ - 12x² - 10x + 8

5. What is the composition of functions denoted by f ⊙ g?
A) f(g)
B) g(f)
C) f(g(x))
D) g(f(x))

6. Which of the following represents the domain of the sum of two functions f and g?
A) Dom(f ∪ g)
B) Dom(f) ∩ Dom(g)
C) Dom(fg)
D) Dom(f - g)

7. Two functions have the same range but different domains. Are they equal according to the defined equality of functions?
A) Yes
B) No
C) Depends on the specific functions
D) Insufficient information provided

8. What is the result of (f * g)(x) if f(x) = x + 1 and g(x) = 2x?
A) 2x² + 2x
B) 2x² + 1
C) 2x² + 3x
D) 3x² + 2x

9. Given f = {(2, z), (3, q)} and g = {(a, 2), (b, 3), (c, 5)}, what is the composition (f ⊙ g)(x)?
A) {(a, z), (b, z), (c, z)}
B) {(2, a), (3, b)}
C) {(2, 3), (3, 5)}
D) Cannot be determined

10. The domain of (f + g)(x) is the intersection of:
A) Dom(f) and Dom(g)
B) Dom(f) and Range(g)
C) Dom(f) and Codomain(g)
D) Range(f) and Dom(g)

11. Which function is a real-valued function?
A) h(x) = √x
B) k(x) = 1/x
C) m(x) = 4x^2 + 3x
D) n(x) = log(x)

12. What does the composition of functions f ⊙ g indicate?
A) g followed by f
B) f followed by g
C) Division of f by g
D) Taking the square root of f

13. If the functions have different domains but the same range, can they be considered equal?
A) Yes
B) No
C) Depends on specific functions
D) Need more information to decide

14. Find the difference of functions f(x) = 2x^2 - x and g(x) = x^2 - 3.
A) x^2 - 2x + 3
B) x^2 - x + 3
C) x^2 - 4x - 3
D) x^2 + 2x - 3

15. What additional condition is necessary to define the quotient of two functions?
A) g(x) should have a maximum value.
B) g(x) must be positive.
C) g(x) must not equal zero.
D) f(x) should be an even function.

16. Which operation is defined between two functions f and g as (f * g)(x)?
A) Addition
B) Subtraction
C) Multiplication
D) Division

17. Determine the result of (f + g)(x) if f(x) = x^2 - 3x and g(x) = 2x.
A) x^2 - x
B) x^2 + x
C) 3x^2 - 2x
D) 3x^2 + x

18. For the functions f(x) = x^3 + 1 and g(x) = 2x, find (f - g)(x).
A) x^3 - 1
B) x^3 + 2x - 1
C) x^3 - 2x - 1
D) x^3 + 1 - 2x

19. If f(x) = 4x + 2 and g(x) = x^2, what is the result of (f * g)(x)?
A) 4x^3 + 2x
B) 4x^3 + 2x^2
C) 4x^2 + 2x
D) 4x^2 + x^2

20. Identify the domain of (f * g)(x) given f(x) = x^2 and g(x) = 3x - 1.
A) All real numbers
B) x ≠ 1/3
C) x ≠ 0
D) x ≠ 1
Answer
https://t.me/yohana1234567/1090

Types of functions and inverse of a function

1. What defines a one-to-one function?
A) For all x1, x2 ∈ A, if f(x1) = f(x2), then x1 ≠ x2
B) Every element of B is an image of some element in A
C) Range(f) = B
D) f is both 1-1 and onto

2. An onto function is characterized by:
A) Each element of A is mapped to one element of B
B) Every element of B is an image of some element in A
C) No two elements of A are mapped to the same element in B
D) f is a correspondence

3. What is the definition of a 1-1 correspondence?
A) f is both 1-1 and onto
B) f is a one-to-one function
C) Range(f) = B
D) Each element of A is mapped to exactly one element of B

4. How is the inverse of a function denoted?
A) f'
B) f*
C) f^(-1)
D) inv(f)

5. If f = {(2,4), (3,6), (1,7)}, what is f^(-1)?
A) {(4,2), (6,3), (7,1)}
B) {(2,4), (3,6), (1,7)}
C) {(4,2), (6,3), (4,5)}
D) {(4,2), (6,3), (7,1)}

6. Which of the following functions does not have an inverse?
A) f = {(2,4), (3,6), (1,7)}
B) g = {(2,4), (3,6), (5,4)}
C) h = {(2,4), (3,5), (4,6)}
D) All functions have inverses

7. What is the significance of a one-to-one function in finding the inverse?
A) They never have an inverse
B) Only one-to-one functions have inverses
C) They require a special formula to find the inverse
D) They can have multiple inverses

8. How do you determine if a function has an inverse?
A) Check if the function is onto
B) Check if the function is one-to-one
C) Find the range of the function
D) Swap x and y in the function equation

9. In finding the inverse of a one-to-one function, what is the first step?
A) Swap x and y in the equation
B) Swap the domain and co-domain
C) Swap the range and domain
D) Solve for x in terms of y

10. What happens when the inverse of a function is not a function?
A) The function has multiple inverses
B) The function is not one-to-one
C) The function does not have an inverse
D) The function is onto, not one-to-one

11. Which operation is used to determine if a function is onto?
A) Intersection
B) Union
C) Composition
D) Substitution

12. What defines a correspondence in functions?
A) The function is onto
B) The function is one-to-one
C) The function is both one-to-one and onto
D) The function maps some elements of A to multiple elements of B

13. When finding the inverse of a function, why is it crucial to determine if it exists?
A) To optimize efficiency in calculations
B) To ensure the function is onto
C) To meet mathematical conventions
D) To avoid attempting the impossible

14. The domain of the function and its inverse are:
A) Identical
B) Complementary
C) Inverse
D) Interchanged

15. A one-to-one function is said to have an inverse because:
A) It is a rule in mathematics
B) It simplifies function operations
C) Only one-to-one functions satisfy the condition for having an inverse
D) Inverses guarantee bijective functions

16. In a one-to-one function, what does f(x) = f^(-1)(x) imply?
A) Line symmetry
B) One-to-one correspondence
C) Identity function
D) Inverse function

17. For an onto function, what can be said about its range and co-domain?
A) They are always equal
B) They are subsets of each other
C) They are completely unrelated
D) They have a one-to-one correspondence

18. When finding the inverse of a function, what property ensures the uniqueness of the inverse?
A) Onto property
B) One-to-one property
C) Range property
D) Composition property

19. How is the composition of f^(-1) and f related in a bijective function?
A) They are always equal
B) They cancel each other out
C) They form the identity function
D) They create a cyclical function

20. In a one-to-one correspondence, the function:
A) Is not necessarily onto
B) Must be onto
C) Must be one-to-one
D) Has no restrictions
https://t.me/yohana1234567/1091

Polynomials, zeros of polynomials, rational functions and their graphs
PART 1
1. The demand function for a company's product is given by \( p = 80 - x^2 \). This function is an example of a:
A) Linear function
B) Quadratic function
C) Cubic function
D) Quartic function

2. The revenue function \( R = x(80 - x^2) \) is an example of:
A) Linear function
B) Quadratic function
C) Cubic function
D) Quartic function

3. How many zeros can a polynomial of degree \( n \) have at most?
A) n
B) n - 1
C) n + 1
D) n/2

4. If a polynomial function of degree 5 can have at most 4 turning points, how many turning points can a cubic function have at most?
A) 2
B) 3
C) 4
D) 5

5. What is the degree of the polynomial function \( p(x) = 2x^2 + 1 \)?
A) 1
B) 2
C) 3
D) 4

6. Which theorem states that if \( a + bi \) is a zero of a polynomial, then \( a - bi \) is also a zero?
A) Conjugate Roots Theorem
B) Rational Root Theorem
C) Factor Theorem
D) Remainder Theorem

7. How can one find the zeros of a quadratic function \( Ax^2 + Bx + C \)?
A) Using long division
B) Using the Rational Root Theorem
C) Using the Quadratic formula
D) Using synthetic division

8. The Rational Root Theorem helps find roots of polynomial functions by:
A) Factoring the polynomial
B) Using long division
C) Finding rational solutions
D) Simplifying complex roots

9. A rational function is a function of the form:
A) \( f(x) = \frac{a(x)}{b(x)} \) where \( a(x) \) is a polynomial
B) \( f(x) = a(x)b(x) \) where \( a(x) \) is a polynomial
C) \( f(x) = a(x) \) where \( a(x) \) is a polynomial
D) \( f(x) = a(x) + b(x) \) where \( a(x) \) and \( b(x) \) are polynomials

10. Which of the following is an example of a rational function?
A) \( f(x) = 3x^2 + 5 \)
B) \( f(x) = \frac{1}{x} \)
C) \( f(x) = \sqrt{x} \)
D) \( f(x) = x + 1 \)

11. The domain of a rational function is given by:
A) {x : d(x) = 0}
B) {x : b(x) ≠ 0}
C) {x : b(x) = 0}
D) {x : d(x) ≠ 0}

12. How do you find the domain of the function \( f(x) = \frac{x^2 - x - 12}{x} \)?
A) By setting the denominator equal to zero
B) By solving the quadratic equation in the numerator
C) By using long division
D) By finding the zeros of the function

13. What are the zeros of the function \( f(x) = x^2 - x - 12 \)?
A) 3, -4
B) -3, 4
C) 3, 4
D) -3, -4

14. The Rational Root Theorem involves finding rational solutions for polynomial functions with:
A) Imaginary coefficients
B) Equal coefficients
C) Integer coefficients
D) Prime coefficients

15. If a polynomial has a rational root \(\frac{p}{q}\), then p is a factor of:
A) Leading coefficient of the polynomial
B) Constant term of the polynomial
C) Sum of coefficients of the polynomial
D) Product of coefficients of the polynomial

16. The Conjugate Roots Theorem states that if a + bi is a zero of a polynomial, then:
A) a - bi is also a zero
B) a bi is also a zero
C) a + b is also a zero
D) a b is also a zero

17. How many turning points can a polynomial of degree 4 have at most?
A) 1
B) 2
C) 3
D) 4

18. How can the Rational Root Theorem help in finding zeros of higher degree polynomials?
A) By providing a method to factorize the polynomials
B) By narrowing down potential rational solutions
C) By guaranteeing the existence of rational roots
D) By simplifying the division process

19. Given a polynomial \( r(x) = x^4 + 2x^3 - 9x^2 + 26x - 20 \), if \( 1 - 3i \) is a zero, what is the other zero of \( r(x) \)?
A) 1 + 3i
B) -1 - 3i
C) 1 - 3i
D) -1 + 3i

20. How many zeros can a polynomial of degree 6 have at most?
A) 5
B) 6
C) 7
D) 8
Answer
https://t.me/yohana1234567/1092

Part 2
1. What is the leading coefficient of a polynomial function?
A) the coefficient of the term with the highest exponent
B) the constant term in the polynomial
C) the term with the lowest exponent
D) the coefficient of the quadratic term

2. Which of the following is true about a polynomial of degree 1?
A) It has at most one zero
B) It has exactly one zero
C) It has at most two zeros
D) It has an infinite number of zeros

3. What does the Location Theorem state about zeros of a polynomial function?
A) It guarantees the existence of at least one zero within an interval
B) It provides a method to factorize polynomials
C) It predicts the exact location of all zeros
D) It specifies the domain of the polynomial function

4. The Rational Root Theorem states that if q/p is a rational root of a polynomial function, then:
A) q is a factor of an and p is a factor of a0
B) q is a factor of a0 and p is a factor of an
C) q is a factor of the constant term and p is a factor of the leading coefficient
D) q is a factor of the leading coefficient and p is a factor of the constant term

5. The Fundamental Theorem of Algebra states that:
A) Every polynomial has at least one real root
B) Every polynomial has at least one complex root
C) Every polynomial has at least one imaginary root
D) Every polynomial has at least one rational root

6. Which theorem allows us to recognize the relationship between factors of a polynomial and its zeros?
A) Factor Theorem
B) Remainder Theorem
C) Conjugate Roots Theorem
D) Location Theorem

7. The Factor Theorem states that x - r is a factor of a polynomial p(x) if:
A) p(r) = 0
B) p'(x) = r
C) p''(r) = 0
D) p(r) = 1

8. How do you find the zeros of a polynomial using the Remainder Theorem?
A) By checking if the remainder is greater than zero
B) By substituting x = 0 into the polynomial
C) By dividing the polynomial by (x - r) and evaluating p(r)
D) By differentiating the polynomial

9. According to the Conjugate Roots Theorem, if 1 - 3i is a zero of a polynomial, what must also be a zero?
A) 1 + 3i
B) 1 + 2i
C) 1 - 2i
D) 1 - i

10. What does the Factor Theorem establish regarding zeros of a polynomial?
A) The relation between factors of a polynomial and its zeros
B) The existence of only real roots for a polynomial
C) The exclusivity of complex roots for a polynomial
D) The dominance of irrational roots over rational roots

11. The Rational Function (f(x) = n(x)/d(x)) is defined as:
A) a function expressed as the ratio of two polynomials
B) a function with only linear terms
C) a function with a constant denominator
D) a function with a quadratic numerator

12. In a polynomial of degree n, how many zeros can it have at most?
A) n
B) n - 1
C) n + 1
D) Two

13. The Linear Factorization Theorem states that a polynomial of degree n can be expressed as:
A) n linear factors
B) n quadratic factors
C) n cubic factors
D) n quartic factors

14. What is the Domain of a rational function f(x) = n(x)/d(x)?
A) {x : d(x) ≠ 0}
B) {x : n(x) ≠ 0}
C) {x : f(x) ≠ 0}
D) {x : n(x) = 0}

15. How many zeros does a polynomial of degree 3 have, as per the Linear Factorization Theorem?
A) 1
B) 2
C) 3
D) 4

16. The Remainder Theorem is based on the divisibility of:
A) p'(x)
B) (x - r)
C) q'(x)
D) (x + r)

17. What is the standard form of a polynomial function of degree 3?
A) an x^3 + an-1 x^2 + ... + a0
B) ax^3 + bx^2 + cx + d
C) (x - r1)(x - r2)(x - r3)
D) a3 x^3 + a2 x^2 + a1 x + a0

18. How do you find the remainder when p(x) = x^3 - x^2 + 3x - 1 is divided by x - 2?
A) p(2)
B) p'(2)
C) p(3)
D) p'(3)

19. According to the Factor Theorem, x - r is a factor of p(x) if:
A) p(r) = 0
B) p(r) = 1
C) p'(r) = 0
D) p''(r) = 0

20. What theorem can be used to find at least one zero of a polynomial within a specified interval?
A) Conjugate Roots Theorem
B) Location Theorem
C) Rational Root Theorem
D) Factor Theorem

Answer
1. B) The leading coefficient
2. B) Quadratic function
3. B) q(x) = 3x^4 + 2x - π
4. D) Forms a smooth unbroken curve
5. B) 2
6. C) 3
7. D) Increasing y-values
8. D) The roots of the polynomial equation p(x) = 0
9. D) The remainder after division
10. C) The roots or zeros of p(x)
11. C) r is a root of the polynomial equation p(x) = 0
12. B) The weights given to different terms
13. B) One less than its degree
14. C) 2
15. C) By solving the equation p(x) = 0
16. B) The end behavior of the function
17. C) Quintic function
18. A) Determines the location of the origin on the graph
19. B) At most n - 1 turning points
20. C) The domain of a polynomial function is always the set of real numbers

Economic final Ddu
1, https://t.me/tumimfamil/391?single
2, https://t.me/tumimfamil/407?single
3, https://t.me/tumimfamil/419?single

Chersachu simut

💞💞💞በምክንያት ነዉ
ትላንትናችን ላይ የተደረጉት ነገሮች ዛሬያችን ላይ ሊያስደረጉን የሚችሉበትን አቅም በኛ ይወልዳሉ አንዳንድ ምክንያቶች ትላንታችን ላይ የተደረጉበት ሁነኛ ሚስጥር ዛሬያችን ላይ ለምናደርጋቸው ብቃት መወለጃ ናቸው።
👦ሰዉ ሆይ ምክንያቶች ያለ ምክንያት አይኖሩም
ተደራግ ምክንያቶች ጀርባቸው ላይ አስደረጊ ምክንያቶች አላቸው ዛሬ ለምናደርጋቸው ምክንያቶች አስደናቂ ምክንያቶቻችን ትዝታ ውስጥ ገብተን ልናደርግ የቻልነው። በኛነታችን በስብእናችን በፍላጎታችን ሳይሆን ትላንታችንን የተቀላቀለው ምክንያት ዛሬያችን ላይ የምናደርጋቸውን ምክንያቶች  ወልዷል ስለዚህ ምክንያቶች ምክንያት አላቸው።
ኃላ የለለዉ ፍት የለዉም
ትላንት የለለዉ ዛሬ የለዉም
ለዝህም መፅሐፍ ቅዱስ
ሲናገር የተቆረጥክበትን ድንጋይ ደግሞ የተማስክበትን ጉድጓድ ወደ ኋላ ሄደህ አባትህን አብርሃም እናትህን ሳራ  ተመልከት ይላል። ትላንት ካላቹ ነገ አላቹ  ስለዚህ ብዙ ምክንያቶች የህይወት አለማችን ላይ ተቀዛቅዘዉብን ከሆነ ልንሆን የምንፈልጋቸውን ነገሮች ባለመሆን ከድነናቸው ከሆነ መሯሯጥ የምንችልባቸውን አቅሞች ወደ ዳር ጥለናቸው ከሆነ
ትላንትናችን ላይ የተሯራጠልን ለላ አቅም እንዳለ እየረሳን ነው ማለት ነው።የሕይወት ዓለማችን በድብዛዘ ከተሞላ
ዛሬ ጎላጎላ እንዲል ወደ ትላንትና እናያለን ዛር ብለን የምናየው ቀና ብለን እንድናይ አቅም ይወልድልናል።
ትላንት አናልፍም ብለን ተስፋ የቆረጥን የተከፋን  ቀን እና ለልቶች  ነበሩ:: የእግዚአብሔር ክንድ 💪💪  ብርታታችን ሆኖ ያለፍናቸዉ::
እና ዛሬ የደከመ ከመሰላቹ ወደ ትላንትናቹ እዩ
አሁን ላይ በራሳችን ልንሰራ እየሞከርን ይሆናል
ለሰው የማይቻል ለእግዚአብሔር ይቻላል።

ራእይ 2
4 የቀደመውን ፍቅርህን ትተሃልና።

Economics