Question

Suppose that the functions u and w are defined as follows :

Answer 1

Answer:

(u o w) (7) =  22

(w o u) 7) = 8

Step-by-step explanation:

We are given:

$u(x)= x^2+6\\w(x)= \sqrt{x+9}$

We need to find:

a) (u o w) (7)

First we will find (u o w) (x) and then we will find (u o w) (7)

We know that (u o w) (x) = u(w(x))

Put value of w(x) into u(x)

we have:

$u(x)=x^2+6\\Put\: x =\sqrt{x+9}\\u(w(x))=(\sqrt{x+9})^2+6\\u(w(x))=x+9+6\\ u(w(x))=x+15$

Now finding (u o w) (7)

We know that: (u o w) (7) = u(w(7))

$u(w(x))=x+15\\Put\:x=7\\u(w(7))=7+15\\u(w(7))=22$

So, (u o w) (7) = 22

b) (w o u) (7)

First we will find (w o u) (x) and then we will find (w o u) (7)

We know that (w o u) (x) = w(u(x))

Put value of u(x) into w(x)

we have:

$w(x)= \sqrt{x+9}\\Put\:x=x^2+6\\w(u(x))= \sqrt{(x^2+6)+9}\\w(u(x))= \sqrt{x^2+6+9}\\w(u(x))= \sqrt{x^2+15}$

Now finding (w o u) (7)

We know that (w o u) (7) = w(u(7))

$w(u(x))= \sqrt{x^2+15}\\Put\:x=7\\w(u(7))= \sqrt{(7)^2+15}\\w(u(7))= \sqrt{49+15}\\w(u(7))= \sqrt{64}\\w(u(7))= 8$

So, (w o u) (7) = 8