IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Join our knowledgeable community to find the answers you need for any topic or issue.

Complete the ordered pairs that lie on the graph of the function [tex]h[/tex].

[tex]
h(x)=\left\{\begin{array}{ll}
-x^2 - 6x - 9, & x \ \textless \ -2 \\
\left(\frac{1}{3}\right)^x - 4, & -2 \leq x \leq 2 \\
\frac{1}{2} x - 4, & x \ \textgreater \ 2
\end{array}\right.
[/tex]

Type the correct answer in each box. Use numerals instead of words.

(-4, [tex]\square[/tex])

(0, [tex]\square[/tex])

(8, [tex]\square[/tex])


Sagot :

We need to find the values of [tex]\(h(x)\)[/tex] for the given [tex]\(x\)[/tex]-coordinates and complete the ordered pairs.

1. For [tex]\( x = -4 \)[/tex]:
Since [tex]\( -4 < -2 \)[/tex], we use the first piece of the function:
[tex]\[ h(x) = -x^2 - 6x - 9 \][/tex]
Plugging in [tex]\( x = -4 \)[/tex]:
[tex]\[ h(-4) = -(-4)^2 - 6(-4) - 9 \][/tex]
[tex]\[ h(-4) = -16 + 24 - 9 \][/tex]
[tex]\[ h(-4) = -1 \][/tex]

So, the ordered pair is [tex]\( (-4, -1) \)[/tex].

2. For [tex]\( x = 0 \)[/tex]:
Since [tex]\(-2 \leq 0 \leq 2\)[/tex], we use the second piece of the function:
[tex]\[ h(x) = \left( \frac{1}{3} \right)^x - 4 \][/tex]
Plugging in [tex]\( x = 0 \)[/tex]:
[tex]\[ h(0) = \left( \frac{1}{3} \right)^0 - 4 \][/tex]
[tex]\[ h(0) = 1 - 4 \][/tex]
[tex]\[ h(0) = -3 \][/tex]

So, the ordered pair is [tex]\( (0, -3) \)[/tex].

3. For [tex]\( x = 8 \)[/tex]:
Since [tex]\( 8 > 2 \)[/tex], we use the third piece of the function:
[tex]\[ h(x) = \frac{1}{2}x - 4 \][/tex]
Plugging in [tex]\( x = 8 \)[/tex]:
[tex]\[ h(8) = \frac{1}{2}(8) - 4 \][/tex]
[tex]\[ h(8) = 4 - 4 \][/tex]
[tex]\[ h(8) = 0 \][/tex]

So, the ordered pair is [tex]\( (8, 0) \)[/tex].

Thus, completing the ordered pairs:
[tex]\[ (-4, \mathbf{-1}),\ (0, \mathbf{-3}),\ (8, \mathbf{0}) \][/tex]