Question

Let f={(-1,-8),(1,-2),(2,1),..} be a function from Z to Z defined by f(x)=px+q for some integers p and q determine p and q

Answer 1

Step-by-step explanation:

f={(1,1),(2,3),(0,−1),(−1,−3)}

f(x)=ax+b

(1,1)∈f

⇒f(1)=1

⇒a×1+b=1

⇒a+b=1 .

(0,−1)∈f

⇒f(0)=−1

⇒a×0+b=−1

⇒b=−1

On substituting b=−1 in eqn (1), we get

a+(−1)=1

⇒a=1+1=2

Thus the respective values of a and b are 2 and −1

Answer 2

The value of p = 3 and q = - 5

Given :

Let f = {(-1,-8) , (1,-2) , (2,1) , ..} be a function from Z to Z defined by f(x) = px + q for some integers p and q

To find :

The value of p and q

Solution :

Step 1 of 2 :

Form the equation to find the value of p and q

Here the given function is

A function from Z to Z defined by f(x) = px + q for some integers p and q

f = {(-1,-8) , (1,-2) , (2,1) , ..}

$\displaystyle \sf \therefore \: f( - 1) = - 8 \: \: \: and \: \: \: f(1) = - 2$

Now ,

$\displaystyle \sf f( - 1) = - 8 \: \: gives$

$\displaystyle \sf{ \implies }p \times ( - 1) + q = - 8$

$\displaystyle \sf{ \implies } - p + q = - 8 \: \: \: - - - - (1)$

Again ,

$\displaystyle \sf f( 1) = - 2 \: \: gives$

$\displaystyle \sf{ \implies }(p \times 1) + q = - 2$

$\displaystyle \sf{ \implies } p + q = - 2 \: \: \: - - - - (2)$

Step 2 of 2 :

Find the value of p and q

$\displaystyle \sf{ } - p + q = - 8 \: \: \: - - - - (1)$

$\displaystyle \sf{ } p + q = - 2 \: \: \: - - - - (2)$

Adding Equation 1 and Equation 2 we get

$\displaystyle \sf{2q = - 10 }$

$\displaystyle \sf{ \implies }q = - 5$

From Equation 2 we get

$\displaystyle \sf p + ( - 5) = - 2$

$\displaystyle \sf{ \implies }p - 5 = - 2$

$\displaystyle \sf{ \implies }p = - 2 + 5$

$\displaystyle \sf{ \implies }p = 3$

Hence value of p = 3 and q = - 5

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