week 5 math 230

Problem (1)

A)

The function f(x)=x2 and g(x)=|x|2 are equal.

To arrive at this conclusion, note that for any integer x, the absolute value |x| satisfies |x|2=x2 because squaring a number yields the same result regardless of its sign. For example: If x=3, then f(3)=9 and g(3)=|3|2=9. If x=-3, then f*(-3)=9 and g*(-3)=|-3|2=9. If x=0, then f(0)=0 and g(0)=|0|2=0. This holds for all integers, so f(x)=g(x) everywhere on the domain mathbb(Z). b The functions f:*mathbb(Z)×mathbb(Z)*to*mathbb(Z) where f(x,y)=|x|+|y|

B)

The functions f:*mathbb(Z)×mathbb(Z)*to*mathbb(Z) where f(x,y)=|x|+|y| and g:*mathbb(Z)×mathbb(Z)*to*mathbb(Z) where g(x,y)=|x+y| are not equal. An element of the domain where they differ is (1-1), since f(1,-1)=|1|+|-1|=2 but g(1,-1)=|1+(-1)|=0. To arrive at this conclusion, compare the expressions: f(x,y) sums the absolute values, which is always non-negative and measures the "Manhattan distance" from (0*0). g(x,y) takes the absolute value of the sum, which can be smaller if x and y have opposite signs. Testing specific points reveals the difference. For the chosen point: f(1,-1)=1+1=2. g(1,-1)=|0|=0. This mismatch confirms they are not equal, as functions must agree on all inputs to be equal.

Problem 2

this is more of a play into how are how functions are ordered we need to pay attention to the fact that this is a function yes and then the x is assigned to it.

c and e are equal, as both are the function h(x)=(x-17)/10. To arrive at this solution, first compute each composition explicitly:

a (f○g)*(x)=f(g(x)*(x))=f*(5x+7)=2*(5x+7)+3=10x+17

b (g○f)*(x)=g(f(x)*(x))=g*(2x+3)=5*(2x+3)+7=10x+22

after a and b our composition as shown in c d e has a negatvive exponet above our function, we can figure out if the are similar by first finding the actual inverse of our orginal function and comparing these functions so for our orginal functions our inverses are

for f we have f(-1)*(x)=(x-3)/2 for g we have g(-1)*(x)=(x-7)/5

next we are comparing our functions for relationships to see if they are a composition.

c (f○g)(-1)*(x)=(x-17)/10 inverse of 10x+17

d (f(-1)○g(-1))*(x)=f(-1)*(g(-1)*(x))=f(-1)*((x-7)/5)=((x-7)/5-3)/2=(x-22)/10

e (g(-1)○f(-1))*(x)=g(-1)*(f(-1)*(x))=g(-1)*((x-3)/2)=((x-3)/2-7)/5=(x-17)/10 Comparing these, c equals e. This follows the property that (f○g)(-1)=g(-1)○f(-1). No other pairs are equal.