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Alborosie
2 years ago
13

Slope-intercept equation from graph

Mathematics
1 answer:
kicyunya [14]2 years ago
5 0
The answer is y=4x+-9

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Which of the following functions are homomorphisms?
Vikentia [17]
Part A:

Given f:Z \rightarrow Z, defined by f(x)=-x

f(x+y)=-(x+y)=-x-y \\  \\ f(x)+f(y)=-x+(-y)=-x-y

but

f(xy)=-xy \\  \\ f(x)\cdot f(y)=-x\cdot-y=xy

Since, f(xy) ≠ f(x)f(y)

Therefore, the function is not a homomorphism.



Part B:

Given f:Z_2 \rightarrow Z_2, defined by f(x)=-x

Note that in Z_2, -1 = 1 and f(0) = 0 and f(1) = -1 = 1, so we can also use the formular f(x)=x

f(x+y)=x+y \\  \\ f(x)+f(y)=x+y

and

f(xy)=xy \\  \\ f(x)\cdot f(y)=xy

Therefore, the function is a homomorphism.



Part C:

Given g:Q\rightarrow Q, defined by g(x)= \frac{1}{x^2+1}

g(x+y)= \frac{1}{(x+y)^2+1} = \frac{1}{x^2+2xy+y^2+1}  \\  \\ g(x)+g(y)= \frac{1}{x^2+1} + \frac{1}{y^2+1} = \frac{y^2+1+x^2+1}{(x^2+1)(y^2+1)} = \frac{x^2+y^2+2}{x^2y^2+x^2+y^2+1}

Since, f(x+y) ≠ f(x) + f(y), therefore, the function is not a homomorphism.



Part D:

Given h:R\rightarrow M(R), defined by h(a)=  \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)

h(a+b)= \left(\begin{array}{cc}-(a+b)&0\\a+b&0\end{array}\right)= \left(\begin{array}{cc}-a-b&0\\a+b&0\end{array}\right) \\  \\ h(a)+h(b)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)+ \left(\begin{array}{cc}-b&0\\b&0\end{array}\right)=\left(\begin{array}{cc}-a-b&0\\a+b&0\end{array}\right)

but

h(ab)= \left(\begin{array}{cc}-ab&0\\ab&0\end{array}\right) \\  \\ h(a)\cdot h(b)= \left(\begin{array}{cc}-a&0\\a&0\end{array}\right)\cdot \left(\begin{array}{cc}-b&0\\b&0\end{array}\right)= \left(\begin{array}{cc}ab&0\\-ab&0\end{array}\right)

Since, h(ab) ≠ h(a)h(b), therefore, the funtion is not a homomorphism.



Part E:

Given f:Z_{12}\rightarrow Z_4, defined by \left([x_{12}]\right)=[x_4], where [u_n] denotes the lass of the integer u in Z_n.

Then, for any [a_{12}],[b_{12}]\in Z_{12}, we have

f\left([a_{12}]+[b_{12}]\right)=f\left([a+b]_{12}\right) \\  \\ =[a+b]_4=[a]_4+[b]_4=f\left([a]_{12}\right)+f\left([b]_{12}\right)

and

f\left([a_{12}][b_{12}]\right)=f\left([ab]_{12}\right) \\ \\ =[ab]_4=[a]_4[b]_4=f\left([a]_{12}\right)f\left([b]_{12}\right)

Therefore, the function is a homomorphism.
7 0
3 years ago
Jeff uses the expression 2(3b + 1) to determine how many books (b) he needs to order for his store. If b = 12, how many books sh
DENIUS [597]
3 x 12 would equal 36, +1 would make it 37. Since there is no space in between 2 and the (, you multiply it. 37 x 3 = 111
So, Jeff should order 111 books
3 0
3 years ago
Read 2 more answers
Rewrite 12/17 as a decimal number. Round to the nearest thousandth.
Natasha_Volkova [10]

Answer:

0.706

Step-by-step explanation:

Convert 12/17 to decimal by dividing the numerator by denominator.

12 ÷ 17 = 0.70588.......

The nearest thousandth is the third decimal place.

If we round up: 0.706

If we round down: 0.705

Because the fourth digit is "8", we round up.

If the digit after the one you want to round to is 5 or greater, then you round up. If the digit after is 4 or less, round down..

12 ÷ 17 = 0.70588....... ≈ 0.706

4 0
3 years ago
75 is 128% of what number
Oksana_A [137]

\bold \pink{let \: the \: number \: be \: x}

\mathsf{128\% \times x = 75}

\\  \\  \frac{128}{100}  \times x = 75 \\  \\ x =  \frac{75}{1}  \times  \frac{100}{128}   \\ \\ x =  \frac{7500}{128}

\boxed{x = 58.59375}

75 is 128% of 58.59375

5 0
3 years ago
Read 2 more answers
Don’t know the answer to this question???
Eva8 [605]

9514 1404 393

Answer:

  (b)  9

Step-by-step explanation:

The product of segment lengths to the near and far intersection with the circle is the same for both secants.

  4(4+5) = 3(3+x) . . . . the above statement with numbers filled in

  36 = 9 +3x . . . . . . . simplify

  12 = 3 +x . . . . . . . . divide by 3

  9 = x . . . . . . . . . . . subtract 3

3 0
3 years ago
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