(A) Given the function $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ defined by $f(x) = ax + b$.Using the point $(0, -1) \in f$,we have $f(0) = -1$.Subs

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(A) Given the function $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ defined by $f(x) = ax + b$.Using the point $(0, -1) \in f$,we have $f(0) = -1$.Substituting $x = 0$ into $f(x) = ax + b$,we get $a(0) + b = -1$,which implies $b = -1$.Using the point $(1, 1) \in f$,we have $f(1) = 1$.Substituting $x = 1$ and $b = -1$ into $f(x) = ax + b$,we get $a(1) + (-1) = 1$.$a - 1 = 1 \Rightarrow a = 2$.Thus,the values are $a = 2$ and $b = -1$.

Class 12 Mathematics Relation and Function Functional Equations

Medium

Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$ for some integers $a$ and $b$. Determine the values of $a$ and $b$.

A
$a = 2, b = -1$
B
$a = 1, b = 2$
C
$a = -2, b = 1$
D
$a = 2, b = 1$

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Relation and Function

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Functional Equations

$f: R \rightarrow R$ is a function such that $f(0)=1$ and for all $x, y \in R$,$f(xy+1)=f(x)f(y)-f(y)-x+2$. Then $\frac{df}{dx}$ at $x=e$ is:

MediumAP EAMCET 2018

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Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy, \forall x, y \in \mathbb{R}$,then $\sum_{n=1}^{10} f(n)$ is equal to:

DifficultAP EAMCET 2022

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Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2x+1}{3}$. If $\alpha$ is an element in the domain of $f$ whose image is $\frac{1}{\alpha}$,then the sum of all possible values of such $\alpha$ is

EasyAP EAMCET 2023

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Exactly how many functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ exist such that $f(x+y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$ for all $x, y \in \mathbb{Q}$?

EasyAP EAMCET 2020

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If $f(x)$ satisfies $f(7 - x) = f(7 + x)$ for all $x \in R$ such that $f(x)$ has exactly $5$ real roots which are all distinct,and the sum of the real roots is $S$,then $S/7$ is equal to:

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Let $f = \{(1, 1), (2, 3), (0, -1), (-1, -3)\}$ be a function from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $f(x) = ax + b$ for some integers $a$ and $b$. Determine the values of $a$ and $b$.

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$f: R \rightarrow R$ is a function such that $f(0)=1$ and for all $x, y \in R$,$f(xy+1)=f(x)f(y)-f(y)-x+2$. Then $\frac{df}{dx}$ at $x=e$ is:

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy, \forall x, y \in \mathbb{R}$,then $\sum_{n=1}^{10} f(n)$ is equal to:

Let $f: R \rightarrow R$ be a function defined by $f(x) = \frac{2x+1}{3}$. If $\alpha$ is an element in the domain of $f$ whose image is $\frac{1}{\alpha}$,then the sum of all possible values of such $\alpha$ is

Exactly how many functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ exist such that $f(x+y) = f(x) + f(y)$ and $f(xy) = f(x)f(y)$ for all $x, y \in \mathbb{Q}$?

If $f(x)$ satisfies $f(7 - x) = f(7 + x)$ for all $x \in R$ such that $f(x)$ has exactly $5$ real roots which are all distinct,and the sum of the real roots is $S$,then $S/7$ is equal to:

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