Question

If we have two functions as $f\left( x \right)=ax+b,g\left( x \right)=cx+d$, then $f\left\{ g\left( x \right) \right\}=g\left\{ f\left( x \right) \right\}$ is equivalent to:
1. $f\left( a \right)=g\left( c \right)$
2. $f\left( b \right)=g\left( b \right)$
3. $f\left( d \right)=g\left( b \right)$
4. $f\left( c \right)=g\left( a \right)$

Answer 1

Hint: For solving this question you should know about the functions and the properties of functions. While solving this question we will put the value of another given function in the other required function. And then we will solve it by expanding that function. And finally we will solve this for a single function.

Complete step-by-step solution:
According to the question, it is asked to us to find the equivalent of $f\left\{ g\left( x \right) \right\}=g\left\{ f\left( x \right) \right\}$ if $f\left( x \right)=ax+b,g\left( x \right)=cx+d$. As we know that the function of any variable is always given in a form that the variable takes at least one term in the entire function. And while we are solving that function, we will put the given values of the given function and solve for that function. Thus we will find the required terms.
According to the question, the functions that are given are,
$f\left( x \right)=ax+b,g\left( x \right)=cx+d$
And the condition that is given is as follows,
$f\left\{ g\left( x \right) \right\}=g\left\{ f\left( x \right) \right\}$
So, now if we put the values according to this given relation, we will get as follows,
$f\left\{ g\left( x \right) \right\}=f\left( cx+d \right)=a\left( cx+d \right)+b$
And we will have the other function as,
$g\left\{ f\left( x \right) \right\}=g\left( ax+b \right)=c\left( ax+b \right)+d$
By this given relation, this can be written as follows,
$\begin{align}
  & acx+ad+b=acx+bc+d \\
 & \Rightarrow f\left( d \right)=g\left( b \right) \\
\end{align}$
Hence the correct answer is option 3.

Note: While solving these types of questions, you have to keep in mind that the values of the functions can be different or similar to each other after solving that function. So, always try to solve the function properly first and then compare it with the other required functions.