CAT 2020 Set-3 | Question: 58

Question

If $ f(x+y) = f(x) f(y) $ and $ f(5) = 4,$ then $ f(10) – f(-10) $ is equal to 

• A. $0$
• B. $15.9375$
• C. $3$
• D. $14.0625$

Answer 1

Given that,

$ f(x+y) = f(x) f(y),$ and $f(5) = 4 $

Now, $ f(10) = f(5+5) $

$ \Rightarrow f(10) = f(5) f(5) $

$ \Rightarrow f(10) = 4 \times 4 $

$ \Rightarrow \boxed{ f(10) = 16} $

And, $ f(15) = f(5+10) $

$ \Rightarrow f(15) = f(5) f(10) $

$ \Rightarrow f(15) = 4 \times 16 $

$ \Rightarrow \boxed{ f(15) = 64} $

We can write $ f(5) = f[15+(-10)] $

$ \Rightarrow f(5) = f(15) f(-10) $

$ \Rightarrow 4 = 64 f(-10) $

$ \Rightarrow  f(-10) = \frac{4}{64}$

$ \Rightarrow \boxed{ f(-10) = \frac{1}{16}} $

Then, $ f(10) – f(-10) = 16 – \frac{1}{16} = \frac{256-1}{16} = \frac{255}{16}  = 15. 9375 $

$ \therefore$ The value of $f(10) – f(-10) = 15 . 9375.$

Correct Answer$: \text{B}$