Question

Statement (A): $f\left( x \right) = \log x$ and $g\left( x \right) = {x^3}$ then $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = f\left[ {g\left( {ab} \right)} \right]$
Statement (B): Every trigonometric function is an even function.
A) Both A and B are true
B) Both A and B are false
C) A is true and B is false
D) A is false and B is true

Answer 1

Here we will use the property of functions $f\left( x \right)$ and $g\left( x \right)$ which states that we can do the composition between these two functions, which means that we can plug $g\left( x \right)$ into $f\left( x \right)$. This is written as $\left( {fog} \right)\left( x \right)$ , pronounced as
$f$ compose $g$ of $x$.
$\left( {fog} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$.

Complete step-by-step answer:
Step (1): For statement (A):
It is given that $f\left( x \right) = \log x$ and $g\left( x \right) = {x^3}$. Now, we need to check if $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = f\left[ {g\left( {ab} \right)} \right]$.
For calculating the LHS side of the equation $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = f\left[ {g\left( {ab} \right)} \right]$, we will substitute the values of functions $f\left( x \right) = \log x$ and $g\left( x \right) = {x^3}$ in it.
$\Rightarrow f\left[ {g\left( a \right)} \right] = \log \left( {{a^3}} \right)$ (\because $$$$g\left( x \right) = {x^3} ) …….. (1)
$\Rightarrow f\left[ {g\left( b \right)} \right] = \log \left( {{b^3}} \right)$ (\because $$$$g\left( x \right) = {x^3} ) …………. (2)
Now, by substituting the values from (1) and (2) in the LHS side of the equation $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = f\left[ {g\left( {ab} \right)} \right]$, we get:
$\Rightarrow f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = \log {a^3} + \log {b^3}$
But we know that $\log {a^3} + \log {b^3} = \log {a^3}{b^3} = \log {\left( {ab} \right)^3}$ , by putting this value in the expression $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right]$, we get:
$\Rightarrow f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = \log {\left( {ab} \right)^3}$
Now for calculating the RHS side of the equation $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = f\left[ {g\left( {ab} \right)} \right]$, we will substitute the values of $f\left( x \right) = \log x$ and $g\left( x \right) = {x^3}$ in it.
$\Rightarrow f\left[ {g\left( {ab} \right)} \right] = \log {\left( {ab} \right)^3}$ (\because $$$$f\left( x \right) = \log x and $g\left( x \right) = {x^3}$)
So, for the expression $f\left[ {g\left( a \right)} \right] + f\left[ {g\left( b \right)} \right] = f\left[ {g\left( {ab} \right)} \right]$, LHS= RHS, hence the statement is true.
Step 2: Statement (2):
We need to check if every trigonometric function is an even function.
Now we know that a function is said to be even if $f\left( { - x} \right) = f\left( x \right)$. For checking this we will take an example as shown below:
For checking if $\sin \left( x \right)$is an even function or not, we need to prove that $\sin \left( { - x} \right) = \sin \left( x \right)$ but that is not true. Because $\sin \left( { - x} \right) = - \sin \left( x \right)$. Hence the function is not even.
It is proved that all trigonometric functions are not even. So, statement (B) is false.

Option C which states that statement A is correct and statement B is false is correct.

Note: In these types of questions, students’ needs to remember that for two different functions $f\left( x \right)$ and $g\left( x \right)$:
$\left( {fog} \right)\left( x \right) = f\left( {g\left( x \right)} \right)$
Also, you should remember that all trigonometric functions are not even. A function is said to be an odd function if for any number $x$ , $f\left( { - x} \right) = - f\left( x \right)$. And a function is said to be even for any number $x$ , $f\left( { - x} \right) = f\left( x \right)$.
Sine and tangent are odd functions. But cosine is an even function.