URGENT – Please show all work, required examples and all calculations.- If you need any explanations, please let me know but the questions should be straight forward.- 1-6 only need to give specific examples when/if the function is not 1-1 or onto- 7-8 – refers to the inverse function- 10 – find the images – … refers tothe set of values actually assumed by the function. The image is not to be confused with the codomain.

CMSC 150 Summer 2016 Section 7380 – Assignment 3

In Problems 1-6, you will be presented with a function and will be asked (a) whether the function

is 1-1, and (b) whether the function in onto. If the function is not 1-1 (or onto) provide a specific

counterexample that shows that the function is not 1-1 (or onto).

Example: f: Z R is defined by f(x) = |x|.

Question

1-1 ?

Onto?

Answer (Yes/No)

No

No

Counterexample (only if No)

f(1) = f(-1)

3.14 is not in the image

Problem 1: g is a function from {1,2,3,4,5} to {5,6,7,8,9}, whose set of ordered pairs is

{(1,8), (2,5), (5,7), (4,9), (3,7)}

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 2: h: R R, h(x) = 23.7x 5.2

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 3: h: R R, h(x) = 1/3x

Question

1-1 ?

Onto?

Answer (Yes/No)

Yes

No

Counterexample (only if No)

Problem 4: k: R R, k(x) = x4

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 5: p: R R, p(x) = 1/(2 + |x | )

Question

1-1 ?

Onto?

Answer (Yes/No)

No

No

Counterexample (only if No)

Problem 6: Let P be the set of all Kings and Queens Regnant of England, and let P0 be the set of

all Kings and Queens Regnant of England except Elizabeth II. Let s represent the successor

function; for example, s(William I) = William II. Note that s is a function from P0 to P.

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 7: Let f: N N be defined by f(n) = 13n 5. Does f -1 exist? If so, find it. If not,

explain why not.

Problem 8: Let g: R R be defined by g(x) = 28 + 7x. Does g -1 exist? If so, find it. If not,

explain why not.

Problem 9: Let f and g be functions from R to R defined by f(x) = 3×2 + 1 and g(x) = x3 – 5.

(a) Find ( g f )( x )

(b) Find ( f g )( x )

Problem 10: Find the images of each of the following functions from R to R.

(a) f ( x ) 2 x 1

(b) f ( x )

1

1 x2

(c) f (x) = 5x

Problem 11: Galileo identified a function from N to N that was 1-1 but not onto. Can you come

up with a function from N to N that is onto but not 1-1?

CMSC 150 Summer 2016 Section 7380 – Assignment 3

In Problems 1-6, you will be presented with a function and will be asked (a) whether the function

is 1-1, and (b) whether the function in onto. If the function is not 1-1 (or onto) provide a specific

counterexample that shows that the function is not 1-1 (or onto).

Example: f: Z R is defined by f(x) = |x|.

Question

1-1 ?

Onto?

Answer (Yes/No)

No

No

Counterexample (only if No)

f(1) = f(-1)

3.14 is not in the image

Problem 1: g is a function from {1,2,3,4,5} to {5,6,7,8,9}, whose set of ordered pairs is

{(1,8), (2,5), (5,7), (4,9), (3,7)}

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 2: h: R R, h(x) = 23.7x 5.2

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 3: h: R R, h(x) = 1/3x

Question

1-1 ?

Onto?

Answer (Yes/No)

Yes

No

Counterexample (only if No)

Problem 4: k: R R, k(x) = x4

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 5: p: R R, p(x) = 1/(2 + |x | )

Question

1-1 ?

Onto?

Answer (Yes/No)

No

No

Counterexample (only if No)

Problem 6: Let P be the set of all Kings and Queens Regnant of England, and let P0 be the set of

all Kings and Queens Regnant of England except Elizabeth II. Let s represent the successor

function; for example, s(William I) = William II. Note that s is a function from P0 to P.

Question

1-1 ?

Onto?

Answer (Yes/No)

Counterexample (only if No)

Problem 7: Let f: N N be defined by f(n) = 13n 5. Does f -1 exist? If so, find it. If not,

explain why not.

Problem 8: Let g: R R be defined by g(x) = 28 + 7x. Does g -1 exist? If so, find it. If not,

explain why not.

Problem 9: Let f and g be functions from R to R defined by f(x) = 3×2 + 1 and g(x) = x3 – 5.

(a) Find ( g f )( x )

(b) Find ( f g )( x )

Problem 10: Find the images of each of the following functions from R to R.

(a) f ( x ) 2 x 1

(b) f ( x )

1

1 x2

(c) f (x) = 5x

Problem 11: Galileo identified a function from N to N that was 1-1 but not onto. Can you come

up with a function from N to N that is onto but not 1-1?

Introduction:

The following is a set of problems related to functions and their properties. These problems are designed to test your understanding of 1-1 and onto functions. The questions will ask you to determine if a given function is 1-1, onto, or both. Additionally, if a function is not 1-1 or onto, you will need to provide a specific counterexample to show why.

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Description:

In this set of problems, you will be presented with different functions and will need to determine if they are 1-1, onto, or both. If a function is not 1-1 or onto, you will need to provide a counterexample to show why. Additionally, there are problems that require you to find the inverse function or the images of certain functions. It is important to note that the image of a function is not the same as the codomain and should not be confused with it. By completing these problems, you will be able to apply your knowledge of 1-1 and onto functions and strengthen your understanding of these fundamental concepts.

Objectives and Learning Outcomes:

I. Determining if a Function is 1-1 or Onto

1. Students will be able to identify if a given function is 1-1 or onto.

2. Students will be able to provide a counterexample to show that a given function is not 1-1 or onto.

II. Understanding Inverse Functions

3. Students will be able to determine if the inverse function of a given function exists.

4. Students will be able to find the inverse function of a given function.

III. Finding Images of Functions

5. Students will be able to find the images of given functions from R to R.

6. Students will be able to differentiate between the image and the codomain of a function.

Headings:

I. 1-1 and Onto Functions

II. Inverse Functions

III. Images of Functions

Note: The specific calculations and examples for each objective have already been provided in the data.

Solution 1:

For Problem 1, the function g is not 1-1 since the inputs 3 and 5 both map to the output 7. The function is also not onto since the output 6 is not mapped to by any input.

Counterexample (only if No):

– Not 1-1: g(3) = g(5) = 7

– Not onto: 6 does not have a corresponding input

For Problem 2, the function h is 1-1 since each input maps to a unique output, but it is not onto since it only covers a portion of the real number line.

Counterexample (only if No):

– Not onto: the output 2 does not have a corresponding input

For Problem 3, the function h is not 1-1 since multiple inputs can map to the same output. The function is also not onto since it does not cover the entire real number line.

Counterexample (only if No):

– Not 1-1: h(-3) = h(3) = 1

– Not onto: the output -1 does not have a corresponding input

For Problem 4, the function k is 1-1 since each input maps to a unique output. It is also onto since every non-negative real number has a fourth root.

For Problem 5, the function p is not 1-1 since inputs with opposite signs map to the same output. The function is also not onto since it approaches 0 as x approaches infinity.

Counterexample (only if No):

– Not 1-1: p(-2) = p(2)

– Not onto: the output 1/2 does not have a corresponding input

For Problem 6, the function s is onto since adding Elizabeth II to the set of inputs allows every member of P to be mapped to via the successor function. However, since Elizabeth II does not have a successor in P0, s is not 1-1.

Counterexample (only if No):

– Not 1-1: s(Elizabeth I) = s(Elizabeth II) = James I

Solution 2:

For Problem 7, the function f is 1-1 since each input maps to a unique output. Its inverse function, f^-1, exists and can be found by solving the equation y = 13x – 5 for x in terms of y, giving x = (y + 5) / 13.

For Problem 8, the function g is 1-1 since each input maps to a unique output. Its inverse function, g^-1, exists and can be found by solving the equation y = 7x – 28 for x in terms of y, giving x = (y + 28) / 7.

For Problem 9,

(a) (g o f)(x) = g(f(x)) = g(3x^2 + 1) = 28 + 7(3x^2 + 1) = 21x^2 + 35

(b) (f o g)(x) = f(g(x)) = f(28 + 7x) = 3(28 + 7x)^2 + 1 = 147x^2 + 1176x + 2350

For Problem 10,

(a) f(x) = 2x + 1, so the image of f is all real numbers.

(b) f(x) = 1/(1 + x^2), so the image of f is all positive real numbers less than or equal to 1.

(c) f(x) = 5x, so the image of f is all real numbers.

For Problem 11, one possible function from N to N that is 1-1 but not onto is f(n) = n + 1. This function never maps to 1, so it is not onto, but it maps every input to a different output, so it is 1-1.

Suggested Resources/Books:

1. “Discrete Mathematics and Its Applications” by Kenneth Rosen

2. “Introduction to Real Analysis” by William Trench

3. “Elementary Differential Equations and Boundary Value Problems” by William Boyce and Richard DiPrima

Similar Asked Questions:

1. What is a 1-1 function?

2. What is an onto function?

3. What is the difference between the image and the codomain of a function?

4. Can a function be both 1-1 and onto?

5. How do you find the inverse of a function?

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