Question

If f(x) is a function such that f(x + y)
f(x)f(y) and f(3) = 125 then f(x) =
1.5
2.x^5
3.5^x
4.5x
​

Answer 1

Question :-

If f(x) is a function such that f(x + y) = f(x)f(y) and f(3) = 125 then f(x) =

1. 5

2. x^5

3. 5^x

4. 5x

$\\ \large\underline{\sf{Solution-}}$

We know that

$\begin{gathered}\:{\underline{\boxed{\bf{{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\sf \: It \: means \: f(x) \: can \: be \: of \: the \: form \: f(x) = {a}^{x}$

$\sf \:  \implies \: f(x + y) = {a}^{x + y}$

$\sf \:  \implies \: f(x + y) = {a}^{x} {a}^{y}$

$\sf \:  \implies \: f(x + y) = f(x) \: f(y)$

$\sf \implies f(x) \: satisfies \: the \: condition \: f(x + y) = f(x)f(y)$

$\sf \: Also, \: it \: is \: given \: that \: f(3) \: = \: 125$

$\sf \:f(3) = 5 \times 5 \times 5$

$\sf \: f(3) \: = \: {5}^{3}$

$\bf\implies \:f(x) \: can \: be \: of \: the \: form \: f(x) = {5}^{x}$

$\large{\boxed{\boxed{\bf{Hence, \: option \: (c) \: is \: correct}}}}$

$\rule{190pt}{2pt}$

Additional information

$\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \dfrac{d}{dx}f(x) \\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf 0 \\ \\ \sf sinx & \sf cosx \\ \\ \sf cosx & \sf - \: sinx \\ \\ \sf tanx & \sf {sec}^{2}x \\ \\ \sf cotx & \sf - {cosec}^{2}x \\ \\ \sf secx & \sf secx \: tanx\\ \\ \sf cosecx & \sf - \: cosecx \: cotx\\ \\ \sf \sqrt{x} & \sf \dfrac{1}{2 \sqrt{x} } \\ \\ \sf logx & \sf \dfrac{1}{x}\\ \\ \sf {x}^{n} & \sf {nx}^{n - 1}\\ \\ \sf {e}^{x} & \sf {e}^{x} \end{array}} \\ \end{gathered}$

Answer 2

Given:- If f(x) is a function such that f(x + y) = f(x)f(y) and f(3) = 125

Find out:- f(x) = ?

According to the question;

as we know

$a^{m} + a^{n} = a^{m+n} \\Now, let f(x) = a^{x} \\f(x+y) = a^{(x+y)} \\f(x+y)= a^{x} a^{y} \\f(x+y)= f(x) f(y)$

now, f(x) satisfies the condition $f(x+y) = f(x) + f(y)$

also,

$f(3) = 125\\f(3)= 5 * 5 * 5\\f(3) = 3^{5}$

Now, $f(x) can be written in the form f(x) = 5^{x}$

Answer is 3. 5^x

If f(x) is a function such that f(x + y)= f(x)f(y) and f(3) = 125 then f(x) = $5^{x}$

Learn More:-

1. function f(x) :- https://brainly.in/question/51668196

2. function f(x)=(x-1):- https://brainly.in/question/16775261

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