Answer to Question #211822 in Discrete Mathematics for Jaguar

Question

Answer questions 4 to 7 by using the given functions g and f. 

Hint: Drawing graphs of f and g before answering the questions, may assist you. Keep in mind 

that g  Z+  Q and f  Z+  Z+. Please note that graphs will not be asked for in the exam.

Question 6

Which one of the following alternatives represents the image of x under g ￮ f (ie g ￮ f(x)))?

1. 20x2 + 8x – 12

2. 80x2 + 4 x –

3. 20x2 + 8x + 3

4. 80x2 + 4 x – 3

Question 7

Which one of the following statements regarding the function g is TRUE? 

(Remember, g  Z+  Q.)

1. g can be presented as a straight line graph.

2. g is injective.

3. g is surjective.

4. g is bijective.

Answer 1

Let $g$ be a function from $Z^+$ (the set of positive integers) to $Q$ (the set of rational numbers) defined by

$$(x, y)\in g\ \text{iff } y=4x-\dfrac{3}{7}\ (g\sube Z^+\times Q)$$

and let $f$ be a function on $Z^+$ defined by

$$(x, y)\in g\ \text{iff } y=5x^2+2x-3\ (f\sube Z^+\times Z^+)$$

Question 6

$$g\circ f(x)=g(5x^2+2x-3)$$

$$=4(5x^2+2x-3)-\dfrac{3}{7}$$

$$=20x^2+8x-12\dfrac{3}{7}$$

$g\circ f:Z^+\to Q$ defined by $g\circ f=20x^2+8x-12\dfrac{3}{7}$

1.  $20x^2+8x-12\dfrac{3}{7}$

Question 7

We consider the statements provided in the different alternatives:

1. $g$ is not defined on the set of real numbers thus $g$ cannot be depicted as a straight line graph.

Only positive integers can be present in the domain of $g.$

It is the case that rdered pairs such as $(1, 3\dfrac{4}{7}), (2, 7\dfrac{4}{7}), (3,11\dfrac{4}{7}),...$ belong to $g$ and these pairs can be presented as dots in a graph. 

2. We prove that $g$ is indeed injective:

$$g(u)=g(v)=>4u-\dfrac{3}{7}=4v-\dfrac{3}{7}$$

$$=>4(u-v)=0=>u=v$$

$g$ is injective.

3. The function $g$ is NOT surjective. 

Counterexample

$$y=\dfrac{1}{2}, \dfrac{1}{2}\in Q$$

Then

$$y=4x-\dfrac{3}{7}=\dfrac{1}{2}=>4x=\dfrac{13}{14}=>x=\dfrac{13}{56}$$

$$\dfrac{13}{56}\not\in Z^+$$

We conclude that $g$ is not surjective. 

4. Since $g$ is not surjective, then $g$ is not bijective.  

Answer

2. $g$ is injective.