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Find “gof” and “fog”where:
\[f(x) = x - 2,\,g(x) = {x^2} + 3x + 1\]?

Last updated date: 17th Mar 2023
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Hint: Here the given question is to find the value of “gof” and “fog” for the given value of the functions, here we know that to obtain the “fog” or “gof”, we need to understand the meaning of “fog” and “gof”, here “fog” means that g(x) function is inside the f(x) function and value of g(x) will be replaced on place of “x” in the “f(x)” function, and respectively for the “gof” also.

Complete step-by-step solution:
Here the given question is to find the value for the given functions, here the property of the “gof” and “fog” is to be used, we already know the meaning of both the functions, now on solving for the values of the functions we get:
First of all dealing with “gof” we get:
\[\Rightarrow gof = {(x - 2)^2} + 3(x - 2) + 1 \\
   \Rightarrow gof = {x^2} - 4x + 4 + 3x - 6 + 1 \\
   \Rightarrow gof = {x^2} - x - 1 \]
Now solving for “fog” we get:
\[ \Rightarrow fog = ({x^2} + 3x + 1) - 2 \\
   \Rightarrow fog = {x^2} + 3x + 1 - 2 = {x^2} + 3x - 1 \]
Here we get the values for the functions.

Note: Here the given question is to solve for the function “fog and gof”, here we can not find the value without getting the meaning of the terms, and as we know that these functions required to solve after putting the value of one function into other and then solve further and get the solution.