Question

Consider the function f and g f(x)= 4x^2 + 1 g(x) x^2-3 perform the function composition

Answer 1

Recall that:

$(h\circ l)(x)=h(l(x)).$

Therefore, to find the composite functions, first, we evaluate f(x) at:

$x^2-3$

and get:

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

Simplifying the above result, we get:

$f(g(x))=4(x^4-6x^2+9)+1=4x^4-24x^2+37.$

Now, we evaluate g(x) at:

$4x^2+1$

and get:

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

Simplifying the above result, we get:

$g(f(x))=16x^4+8x^2-2.$

Answer:

$\begin{gathered} f(g(x))=4x^4-24x^2+37, \\ g(f(x))=16x^4+8x^2-2. \end{gathered}$