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Linear Functions

If f: ℤ→ℤ be ...

Question

If $f : Z \to Z$ be a linear function such that $f (2) + f (1) = 4, f (2) - 2 f (1) = - 1$ , then $f (3)$ is

1

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3

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2

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\frac{2}{3}

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Solution

The correct option is B $3$
We know that linear function can be written as,
$f (x) = a x + b$
Now,
$f (2) = 2 a + b f (1) = a + b$
So,
$f (2) + f (1) = 4 \Rightarrow 2 a + b + a + b = 4 \Rightarrow 3 a + 2 b = 4 \dots (1)$
$f (2) - 2 f (1) = - 1 \Rightarrow 2 a + b - 2 a - 2 b = - 1 \Rightarrow - b = - 1 \Rightarrow b = 1$
Using equation $(1)$ ,
$3 a + 2 \times 1 = 4 \Rightarrow a = \frac{2}{3}$
So,
$f (x) = \frac{2 x}{3} + 1 ∴ f (3) = 2 + 1 = 3$

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Linear Functions

Standard XII Mathematics

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If f:Z→Z be a linear function such that f(2)+f(1)=4, f(2)−2f(1)=−1, then f(3) is

The correct option is B 3We know that linear function can be written as, f(x)=ax+b Now, f(2)=2a+bf(1)=a+b So, f(2)+f(1)=4⇒2a+b+a+b=4⇒3a+2b=4⋯(1) f(2)−2f(1)=−1⇒2a+b−2a−2b=−1⇒−b=−1⇒b=1 Using equation (1), 3a+2×1=4⇒a=23 So, f(x)=2x3+1∴f(3)=2+1=3

If $f : Z \to Z$ be a linear function such that $f (2) + f (1) = 4, f (2) - 2 f (1) = - 1$ , then $f (3)$ is