Question

composition of functions and modeling
decomposing composite functions
the given function ( k ) is a composition of two functions ( m ) and ( n ) so that ( k(x) = (m circ n)(x) ).
( k(x) = \frac{2}{sqrt{4x - 7}} )
if ( n(x) = 4x - 7 ), ( m(x) ) must be equal to which expression?
( 2sqrt4{x} )
( \frac{2}{sqrt{x}} )
( 2sqrt{x} )
( \frac{2}{sqrt4{x}} )

Explanation:

Step1: Recall composite function definition

$(m\circ n)(x) = m(n(x)) = k(x)$

Step2: Substitute $n(x)$ into composite

We know $n(x)=4x-7$, so $m(4x-7) = \frac{2}{\sqrt{4x-7}}$

Step3: Find $m(t)$ by substitution

Let $t = 4x-7$, then $m(t) = \frac{2}{\sqrt{t}}$. Replace $t$ with $x$: $m(x)=\frac{2}{\sqrt{x}}$

Answer:

$\frac{2}{\sqrt{x}}$ (the second option)