Question

If \[f\left( x \right) = 4{x^2} + 3x + 7\]and \[g\left( x \right) = 3x - 5\] then find \[\left( {g \circ f} \right)\left( x \right)\]?

Answer 1

Hint: This question is very easy to solve. We are given two functions separately. And we are asked to find the combination of them that is composite function.What we will do is we will just substitute the value of \[f\left( x \right)\] in the place of $x$ in \[g\left( x \right)\] and solve accordingly. That will be \[\left( {g \circ f} \right)\left( x \right)\].

Complete step by step answer:
Given that, \[f\left( x \right) = 4{x^2} + 3x + 7\] and \[g\left( x \right) = 3x - 5\]
Now what we will do is x term in \[g\left( x \right)\] will be replaced by \[f\left( x \right)\]
\[\left( {g \circ f} \right)\left( x \right) = 3\left( {4{x^2} + 3x + 7} \right) - 5\]
Multiply the terms outside the bracket with the terms inside using distributive property.
\[\left( {g \circ f} \right)\left( x \right) = 12{x^2} + 9x + 21 - 5\]
Since last two terms are constants that can be simplified,
\[\left( {g \circ f} \right)\left( x \right) = 12{x^2} + 9x + 16\]

Therefore, the value of $\left( {g \circ f} \right)\left( x \right) $ is $12{x^2} + 9x + 16$.

Note: The composite function is a combination of two functions but the order of combination definitely matters. What we are learning is \[\left( {g \circ f} \right)\left( x \right)\]is not same as \[\left( {f \circ g} \right)\left( x \right)\] because the function after the small circle is called inner function and the function before the circle is called outer circle. We substitute the value of inner function in the outer function. Do remember the sequence.