Question

Consider the functions f(x) = 2x + 1 and g(x) = 3x^2 - 1. Find and simplify the following:

Answer 1

We are given the following functions:

$\begin{gathered} f\mleft(x\mright)=2x+1,\text{ and} \\ g\mleft(x\mright)=3x^2-1 \end{gathered}$

We are asked to determine the following:

$(f\circ g)(x)$

This is a composition of functions and it is equivalent to the following:

$(f\circ g)(x)=f(g(x))$

This means that we will substitute the value of "x" is f(x) for the function g(x), like this:

$f(g(x))=2(3x^2-1)+1$

Now we simplify, first by applying the distributive property:

$f(g(x))=6x^2-2+1$

Now we solve the operation:

$f(g(x))=6x^2-1$

And thus we have the composition of the functions.

For part B we are asked:

$(g\circ f)(x)=g(f(x))$

This means that this time we will substitute the value of "x" in g(x) for the function f(x), like this:

$g(f(x))=3(2x+1)^2-1$

Now we simplify the function. First, we solve the square using the following relationship:

$(a+b)^2=a^2+2ab+b^2$

Using the relationship we get:

$g(f(x))=3(4x^2+4x+1)-1$

Now we apply the distributive property:

$g(f(x))=12x^2+12x+3-1$

Now we solve the operations:

$g(f(x))=12x^2+12x+2$

And thus we get the composition of the functions.