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Let $f = \left( {1,1} \right),\left( {2,3} \right),\left( {3,5} \right),\left( {4,7} \right)$ be a function $Z$ to $Z$ defined by $f\left( x \right) = ax + b$ for some integers $a,b$. Determine $a,b$.

Answer Verified Verified
Hint: In this question the concept of real valued function i.e. a function whose range is in the real numbers. Example: $f\left( x \right) = x$. Use this to find the value of $a,b$.

Complete step-by-step answer:

According to the question it is given $f = \left( {1,1} \right),\left( {2,3} \right),\left( {3,5}

\right),\left( {4,7} \right)$,

$f\left( x \right) = ax + b.....\left( 1 \right)$

Now, take $f = \left( {1,1} \right)$

Then substituting the values , we get

$

   \Rightarrow f\left( 1 \right) = ax + b \\

   \Rightarrow f\left( 1 \right) = a \times 1 + b \\

   \Rightarrow 1 = a + b \\

   \Rightarrow a + b = 1......\left( 2 \right) \\

$

Now taking $f\left( {2,3} \right)$

$

   \Rightarrow f\left( 3 \right) = 2 \times a + b \\

   \Rightarrow 3 = 2a + b \\

   \Rightarrow 2a + b = 3.........\left( 3 \right) \\

$

On subtracting by equation $\left( 2 \right)$ and $\left( 3 \right)$,

\[

  a + b -2a - b = 1 - 3 \\
 - a = - 2 \\

 \]

$a = 2$ put in equation $\left( 2 \right)$, we get

$

   \Rightarrow b + 2 = 1 \\

   \Rightarrow b = 1 - 2 = - 1 \\

   \Rightarrow b = - 1 \\

 $

Hence $a$ and $b$ will be $2$ and $ - 1$.

Note: In such types of questions where we have found integers of any function so the series of functions that is given to us we can apply the concept of real valued function there and then put the values in the given equation to get the values of $a$ and $b$.