Let f : be a function such that f ( x + y ) = f ( x ) + f ( y ) for all x , y and g : ( 0 ,) be a function such that g ( x + y ) = g ( x ) g ( y ) for all x , y . If f ( - 3 5 ) = 12 and g ( - 1 3 ) = 2 , then the value of ( f ( 1 4 ) + g

Question

Let $f : ℝ \to ℝ$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x, y \in ℝ$ and $g : ℝ \to (0, \infty)$ be a function such that $g (x + y) = g (x) g (y)$ for all $x, y \in ℝ$ . If $f (\frac{- 3}{5}) = 12$ and $g (\frac{- 1}{3}) = 2$ , then the value of $(f (\frac{1}{4}) + g (- 2) - 8) g (0)$ is ________ . [2024]

Smart Achievers · Accepted Answer

(51)

Given $f (x + y) = f (x) + f (y)$

$\Rightarrow f (x) = k x; f (\frac{- 3}{5}) = 12 \Rightarrow k = - 20$ $∴$ $f (x) = - 20 x$

$We have, g (x + y) = g (x) g (y) \Rightarrow g (x) = a^{x}$

$g (\frac{- 1}{3}) = 2 \Rightarrow a = \frac{1}{8}$ $∴$ $g (x) = {(\frac{1}{8})}^{x}$

$(f (\frac{1}{4}) + g (- 2) - 8) g (0) = (- 5 + 64 - 8) \times 1 = 51$

Solution

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Solution

Q. Let f:ℝ→ℝ be a function such that f(x+y)=f(x)+f(y) for all x,y∈ℝ and g:ℝ→(0,∞) be a function such that g(x+y)=g(x)g(y) for all x,y∈ℝ. If f(-35)=12 and g(-13)=2, then the value of (f(14)+g(-2)-8)g(0) is ________ . [2024]

Ans. (51) Given f(x+y)=f(x)+f(y) ⇒f(x)=kx; f(-35)=12⇒k=-20 ∴ f(x)=-20x We have, g(x+y)=g(x)g(y)⇒g(x)=ax g(-13)=2⇒a=18 ∴ g(x)=(18)x (f(14)+g(-2)-8)g(0)=(-5+64-8)×1=51

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