Answered

Suppose that the functions [tex]\( u \)[/tex] and [tex]\( w \)[/tex] are defined as follows:
[tex]\[
\begin{array}{l}
u(x) = x^2 + 5 \\
w(x) = \sqrt{x + 3}
\end{array}
\][/tex]

Find the following:
[tex]\[
\begin{array}{l}
(u \circ w)(1) = \\
(w \circ u)(1) =
\end{array}
\][/tex]



Answer :

GinnyAnswer
Certainly! Let us solve for [tex]\((u \circ w)(1)\)[/tex] and [tex]\((w \circ u)(1)\)[/tex] step by step:

### Finding [tex]\((u \circ w)(1)\)[/tex]:

1. Evaluate [tex]\(w(1)\)[/tex]:
[tex]\( w(x) = \sqrt{x + 3} \)[/tex]
[tex]\[ w(1) = \sqrt{1 + 3} = \sqrt{4} = 2 \][/tex]

2. Substitute [tex]\( w(1) \)[/tex] into [tex]\( u \)[/tex]:
[tex]\( u(x) = x^2 + 5 \)[/tex]
[tex]\[ u(w(1)) = u(2) \][/tex]
[tex]\[ u(2) = 2^2 + 5 = 4 + 5 = 9 \][/tex]

So, [tex]\((u \circ w)(1) = 9\)[/tex].

### Finding [tex]\((w \circ u)(1)\)[/tex]:

1. Evaluate [tex]\(u(1)\)[/tex]:
[tex]\( u(x) = x^2 + 5 \)[/tex]
[tex]\[ u(1) = 1^2 + 5 = 1 + 5 = 6 \][/tex]

2. Substitute [tex]\( u(1) \)[/tex] into [tex]\( w \)[/tex]:
[tex]\( w(x) = \sqrt{x + 3} \)[/tex]
[tex]\[ w(u(1)) = w(6) \][/tex]
[tex]\[ w(6) = \sqrt{6 + 3} = \sqrt{9} = 3 \][/tex]

So, [tex]\((w \circ u)(1) = 3\)[/tex].

Therefore, the required values are:
[tex]\[ (u \circ w)(1) = 9 \][/tex]
[tex]\[ (w \circ u)(1) = 3 \][/tex]

Other Questions