Question

Suppose that the functions u and w are defined as follows: u(x) = x-1w(x) = -2x^2 -2Find the following. (Reference photo)

Answer 1

SOLUTION:

Step 1:

In this question, we have the following:

Step 2:

Part A:

$\begin{gathered} \text{Given }u(x)=x\text{ - 1} \\ \text{and} \\ w(x)=-2x^2\text{ -2} \end{gathered}$

Then, we are to evaluate:

$\text{( w }\circ\text{ u ) (-3) }$

Now, we have that:

$\begin{gathered} (\text{ w }\circ\text{ u ) ( x ) = w ( u ( x ) )} \\ \text{But,} \\ u\text{ ( x) = x - 1} \\ \end{gathered}$$\begin{gathered} \text{But w(x) = -2x}^2\text{ - 2} \\ \text{Now, we are evaluating:} \\ (\text{ w}\circ u)(x)=w(u(x))=w(x-1)=-2(x-1)^2\text{ - 2} \\ \end{gathered}$$\begin{gathered} \text{Then ( w}\circ u\text{ ) ( }-3)=-2(-3-1)^2-2 \\ =-2(-4)^2\text{ - 2} \\ =\text{ -2( 16) -2} \\ =\text{ -32 - 2} \\ =\text{ -34} \end{gathered}$

Then, we have that:

$(\text{ w }\circ u\text{ ) ( - 3) = - 34}$

Part B:

$\begin{gathered} \text{Given ( u}\circ\text{ w) ( x) = u ( w ( x) )} \\ \text{Now, we are evaluating:} \\ (\text{ u}\circ w)\text{ ( - 3)} \end{gathered}$$\begin{gathered} \text{But u ( x) = x - 1} \\ \text{and } \\ w(x)=-2x^2-2 \end{gathered}$$\begin{gathered} u\text{ ( w ( - 3 ) ) = ?} \\ \text{But } \\ \text{u ( w ( x) ) = u ( }-2x^2-2) \\ \text{but u ( x) = x - 1} \end{gathered}$$\begin{gathered} u(-2x^2-2)=(-2x^2-\text{ 2) - 1} \\ \end{gathered}$

But, we need to compare:

$\begin{gathered} -2x^2-2\text{ = -3} \\ -2x^2\text{ = - 3+ 2} \\ -2x^2=\text{ -1} \end{gathered}$

Divide both sides by -2, we have that:

$x^2\text{ = }\frac{1}{2}$

Then,

$\begin{gathered} u(w(-3)=u(-2x^2-2)=(-2x^2\text{ -2) - 1} \\ \end{gathered}$$\text{put x }^2=\text{ }\frac{1}{2},\text{ we have that:}$$\begin{gathered} =\text{ -2 (}\frac{1}{2})^{}\text{ - 2- 1} \\ =\text{ }-1\text{ -2-1} \\ =\text{ -4} \end{gathered}$

CONCLUSION:

We can see that:

$(u\circ w)\text{ ( - 3) = - 4}$