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If $f(x) = \sqrt x $, $x > 0$ and $g(x) = {x^2} - 1$ are two real functions, find $fog$ and $gof$. Is $fog = gof?$

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Last updated date: 16th Jun 2024
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Answer
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Hint: This is a problem which deals with composite functions in mathematics. A composite function is generally a function that is written inside another function. Composite of a function is done by substituting one function into another function. For example, $f\left( {g(x)} \right)$ is the composite function of $f(x)$ and $g(x)$. The composite function $f\left( {g(x)} \right)$ is read as “$f$ of $g$ of $x$”. Similarly $g\left( {f(x)} \right)$ is the composite function of $g(x)$ and $f(x)$. The composite function $g\left( {f(x)} \right)$is read as “$g$ of $f$ of $x$”.

Complete step-by-step answer:
Given the function $f(x) = \sqrt x $, and here given that $x > 0$, which means that the function $f(x)$is only defined for the numbers which are greater than zero. Here $x$ is any number which is greater than zero. And hence $f(x)$ is a real function.
Also given that the function $g(x) = {x^2} - 1$, and here given that $g(x)$ is also a real function.
Here we have to find the composite functions which are $fog$ and $gof$.
Here $fog$ is a composite function, and is defined as the function $f(x)$ of the function $g(x)$ which is a function of $x.$ This composite function is mathematically expressed below:
\[ \Rightarrow fog(x) = f\left( {g(x)} \right)\]
Similarly $gof$ is a composite function, and is defined as the function $g(x)$ of the function \[f(x)\] which is a function of $x.$ This composite function is mathematically expressed below:
\[ \Rightarrow gof(x) = g\left( {f(x)} \right)\]
Now given that $f(x) = \sqrt x $ and $g(x) = {x^2} - 1$, substituting these functions in the composite functions.
Finding the composite function \[fog\] here, as given below:
\[ \Rightarrow fog(x) = f\left( {g(x)} \right)\]
\[ \Rightarrow fog(x) = f\left( {{x^2} - 1} \right)\]
\[ \Rightarrow fog(x) = \sqrt {{x^2} - 1} \]
Here substituting the function $g(x) = {x^2} - 1$, inside the function $f(x)$, and since the function of $f(x)$ is root function, hence anything inside $f(x)$ will be inside the root, as the defined function $f(x)$.
Now finding the composite function \[gof\] here, as given below:
\[ \Rightarrow gof(x) = g\left( {f(x)} \right)\]
\[ \Rightarrow gof(x) = g\left( {\sqrt x } \right)\]
\[ \Rightarrow gof(x) = \sqrt {{{\left( {\sqrt x } \right)}^2} - 1} \]
Here the expression \[{\left( {\sqrt x } \right)^2} = x\], substituting this gives:
\[ \Rightarrow gof(x) = \sqrt {x - 1} \]
Here substituting the function $f(x) = \sqrt x $, inside the function $g(x)$, and since the function of $g(x)$ is an under root function, hence anything inside $g(x)$will be inside the root, as the defined function $g(x)$
Hence the composite functions are obtained as given below:
\[ \Rightarrow fog(x) = \sqrt {{x^2} - 1} \] ;
\[ \Rightarrow gof(x) = \sqrt {x - 1} \]
$\therefore fog \ne gof$

The composite functions are \[fog(x) = \sqrt {{x^2} - 1} \] and \[gof(x) = \sqrt {x - 1} \], here $fog \ne gof$.

Note:
Here while solving this problem please note that a composite function is a function whose values are found from two given functions by applying one function to an independent variable and then applying the second function to the result and whose domain consists of those values of the independent variable for which the result yielded by the first function lies in the domain of the second.