Question

If $f(x) = x + 7$ and $g(x) = x - 7$ , $x \in R.$, then find the value of $f \circ g(6).$

Answer 1

 Hint- Here $f \circ g(6)$ is a composite function. We have to operate $g$ on 6 using the function $g(x) = x - 7$ first and then $f$ will be operated on the result $g(6).$

Complete step by step answer:
In the problem it is given that $g(x) = x - 7.$
We have to substitute 6 for x in order to get$g(6)$, hence we obtain,
$g(6) = 6 - 7 = - 1.$
Generally we know that $f \circ g(x) = f(g(x))$
From the general formula the given composition function can be rewritten as $f \circ g(6) = f(g(6))$
Again, $g(6) = - 1$
So, $f \circ g(6) = f( - 1)$
Let us now substitute -1 for x in$f(x)$, then we get
 $f( - 1) = - 1 + 7 = 6.$
Hence, $f \circ g(6) = 6.$
Additional information: In mathematics composite function is an operation which takes two functions $f$ and $g$ are usually denoted by $g(f(x))$ and produces a function $h$ such that $h(x) = g(f(x)).$ Here $f$ is applied on $x$ first and then $g$ is applied on the result after applying $f$ on $x.$
Suppose $A,B,C$ be three sets and let there be two functions $f$ and $g$ such that $f:A \to B$ and $g:B \to C.$ Then we define the composite of functions $f$ and$g$, which is denoted by $g \circ f$ as $g \circ f:A \to C,$ where $(g \circ f)(x) = g[f(x)],$ for all $x \in A.$
When $f$ is applied to $x$ we have an element of $B.$ Again when $g$ is applied on the result after applying $f$ on $x$ we have an element of $C.$ ultimately when the composite function $g \circ f$ is applied on $x$ in $A$ we get an element on $C.$

Note: The composite functions $f \circ g$ and $g \circ f$ are not the same. In$f \circ g$, $g$ is operated first then f on the result after applying$g$. On the other hand in$g \circ f$, $f$ is operated first then $g$ is operated on the result which is obtained after applying$f$.