Question

Find f o g and g o f, if
f(x) \[={{e}^{x}}\]
g(x) \[={{\log }_{e}}x\]

Answer 1

HINT: -
In mathematics, f o g and g o f are known as composite functions. The function f o g is also represented as f(g(x)) and similarly, function g o f is also represented as g(f(x)).

Complete step-by-step answer:
A composite function is a function that depends on another function. A composite function is created when one function is substituted into another function.
For example, f(g(x)) is the composite function that is formed when g(x) is substituted for x in f(x).
f(g(x)) is read as “f of g of x”.

As mentioned in the question,, we have to find the f o g and g o f for the given functions.
We can do this by using the information that is given in the hint which is as follows
For f o g, we can write it as
\[\begin{align}
  & f\ o\ g={{e}^{{{\log }_{e}}x}} \\
 & f\ o\ g=x \\
\end{align}\]
(By using the property of logarithmic function)
Similarly, for g o f, we can write it as
\[\begin{align}
  & g\ o\ f={{\log }_{e}}({{e}^{x}}) \\
 & g\ o\ f=x \\
\end{align}\]
(By using the property of logarithmic function)
Hence, both f o g and g o f are equal to x.

NOTE: The students can make a mistake in solving this question if they don’t know about logarithmic properties and if they don’t know about the information that is provided in the hint on composite functions.