Question

If $f\left( {x + y} \right) = f\left( x \right) \times f\left( y \right)$and $f\left( 5 \right) = 32\;\;$ then $f\left( {7} \right)$ is equal to
A) $35$
B) $36$
C) $\dfrac{7}{5}$
D) $128$

Answer 1

A function can be explained as a rule which takes each member x of a set and assigns, or maps it to an equivalent value of y known at its image.
x → Function → y
In this problem we will substitute the value of x and y till we get $x + y = 5$. Then try to find the value of $f\left( {1} \right)$. Then using this value we will find the value of $f\left( {7} \right)$.

Complete step-by-step answer:
We have been given in the problem,
$f\left( {x + y} \right) = f\left( x \right) \times f\left( y \right)$ and $f\left( 5 \right) = 32\;\;$
Taking $x{\text{ }} = {\text{ }}y{\text{ }} = {\text{ }}1$ in above equation, we obtain
$f\left( {1 + 1} \right){\text{ = }}f\left( 2 \right) = {\text{ }}f\left( 1 \right)f\left( 1 \right) = {\left[ {f(1)} \right]^2}$
Similarly,
Taking $x{\text{ }} = {\text{ 2}}$ and $y{\text{ }} = {\text{ }}1$ in above equation, we obtain
$f\left( {2 + 1} \right){\text{ = }}f\left( 3 \right) = {\text{ }}f\left( 2 \right)f\left( 1 \right) = {\left[ {f(1)} \right]^3}$
Taking $x{\text{ }} = {\text{ 2}}$ and $y{\text{ }} = {\text{ 2}}$ in above equation, we obtain
$f\left( {2 + 2} \right){\text{ = }}f\left( 4 \right) = {\text{ }}f\left( 2 \right)f\left( 2 \right) = {\left[ {f(1)} \right]^4}$
Taking $x{\text{ }} = {\text{ 2}}$ and $y{\text{ }} = {\text{ 3}}$ in above equation, we obtain
$f\left( {2 + 3} \right){\text{ = }}f\left( 5 \right) = {\text{ }}f\left( 2 \right)f\left( 3 \right) = {\left[ {f(1)} \right]^5}$
Now replacing the value of $f(5)$ from$f\left( 5 \right) = 32\;\;$ in the above equation, we get:
$\Rightarrow f\left( 5 \right) = 32\;\; = {\left[ {f(1)} \right]^5}$
$\Rightarrow {\left[ {f(1)} \right]^5} = {2^5}$
$\Rightarrow f(1) = 2$
We have to find the value of $f\left( {7} \right)$.
As we know 7= 2+5 and we already have a value of $f(5)$ so we will just calculate the value of $f(2)$
As we know ${\text{ }}f\left( 2 \right) = {\left[ {f(1)} \right]^2}$
$\Rightarrow f(2) = {2^2} = 4$
Hence, for finding the value of $f\left( {7} \right)$ using the equation given $f\left( {x + y} \right) = f\left( x \right) \times f\left( y \right)$ we get
$\Rightarrow f(7) = f(5 + 2) = f(5) \times f(2)$
Now substituting the value of $f(5)$ and $f(2)$we get:
$\Rightarrow f(7) = 32 \times 4 = 128$
So, If $f\left( {x + y} \right) = f\left( x \right) \times f\left( y \right)$ and $f\left( 5 \right) = 32\;\;$then $f\left( {7} \right)$ is equal to $128$.

Therefore, the option (D) is the correct answer.

Note: We can solve the question by second method in which we will keep on calculation the values of $f(2),f(3),f(4),f(5),f(6)$ and $f\left( {7} \right)$ in terms of $f(1)$.
Taking $x{\text{ }} = {\text{ 3}}$ and $y{\text{ }} = {\text{ 3}}$ in above equation, we obtain
$f\left( {3 + 3} \right){\text{ = }}f\left( 6 \right) = {\text{ }}f\left( 3 \right)f\left( 3 \right) = {\left[ {f(1)} \right]^6}$
Taking $x{\text{ }} = {\text{ 4}}$ and $y{\text{ }} = {\text{ 3}}$ in above equation, we obtain
$f\left( {4 + 3} \right){\text{ = }}f\left( 7 \right) = {\text{ }}f\left( 4 \right)f\left( 3 \right) = {\left[ {f(1)} \right]^7}$
After this substituting the value of $f(1) = 2$ in the above equation we get:
$\Rightarrow f(7) = {2^7} = 128$