Question

h(x) = \

$$\begin{cases} 6, & -8 \\leq x < -4 \\\\ 3, & -4 \\leq x \\leq 5 \\end{cases}$$

what is the graph of h?
choose 1 answer:
a
\

$$\begin{tikzpicture}scale=0.5 \\draw-> (-9,0) -- (9,0) noderight {$x$}; \\draw-> (0,-9) -- (0,9) nodeabove {$y$}; \\foreach \\x in {-8,-6,-4,-2,2,4,6,8} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\foreach \\y in {-8,-6,-4,-2,2,4,6,8} \\draw (0.1,\\y) -- (-0.1,\\y) nodeleft {\\y}; \\fillblue (-8,6) circle (2pt); \\drawblue (-8,6) -- (-4,6); \\fillwhite (-4,6) circle (2pt); \\fillblue (-4,3) circle (2pt); \\drawblue (-4,3) -- (5,3); \\fillblue (5,3) circle (2pt); \\end{tikzpicture}$$

b
\

$$\begin{tikzpicture}scale=0.5 \\draw-> (-9,0) -- (9,0) noderight {$x$}; \\draw-> (0,-9) -- (0,9) nodeabove {$y$}; \\foreach \\x in {-8,-6,-4,-2,2,4,6,8} \\draw (\\x,0.1) -- (\\x,-0.1) nodebelow {\\x}; \\foreach \\y in {-8,-6,-4,-2,2,4,6,8} \\draw (0.1,\\y) -- (-0.1,\\y) nodeleft {\\y}; \\fillblue (-8,6) circle (2pt); \\drawblue (-8,6) -- (-4,6); \\fillblue (-4,6) circle (2pt); \\fillwhite (-4,3) circle (2pt); \\drawblue (-4,3) -- (5,3); \\fillblue (5,3) circle (2pt); \\end{tikzpicture}$$

Explanation:

Step1: Analyze the first piece of the function

The first piece of the piecewise function is \( h(x) = 6 \) for \( -8 \leq x < -4 \). This means when \( x \) is in the interval from \( -8 \) (inclusive) to \( -4 \) (exclusive), the \( y \)-value is 6. So, at \( x = -8 \), there should be a closed dot (since \( -8 \) is included), and at \( x = -4 \), there should be an open dot (since \( -4 \) is not included) with \( y = 6 \).

Step2: Analyze the second piece of the function

The second piece of the piecewise function is \( h(x) = 3 \) for \( -4 \leq x \leq 5 \). This means when \( x \) is in the interval from \( -4 \) (inclusive) to \( 5 \) (inclusive), the \( y \)-value is 3. So, at \( x = -4 \), there should be a closed dot (since \( -4 \) is included), and at \( x = 5 \), there should be a closed dot (since \( 5 \) is included) with \( y = 3 \).

Step3: Compare with the given graphs

• For option A: The first segment (for \( -8 \leq x < -4 \)) has a closed dot at \( x = -8 \) and an open dot at \( x = -4 \) with \( y = 6 \). The second segment (for \( -4 \leq x \leq 5 \)) has a closed dot at \( x = -4 \) and a closed dot at \( x = 5 \) with \( y = 3 \). This matches our analysis.
• For option B: The first segment has a closed dot at \( x = -4 \) (which is incorrect as \( x = -4 \) is not included in the first interval) and the second segment has an open dot at \( x = -4 \) (which is incorrect as \( x = -4 \) is included in the second interval). So, option B is incorrect.

Answer:

A