Test 11 (Lessons 21–22): Piecewise Functions and Inequalities Solutions

Write the piecewise function for the given graph.

$y = \{\begin{cases} - 2 x + 4 & if x > 4 \\ 3 x + 6 & if x < - 2 \\ \frac{1}{2} x - 3 & if - 1 < x \leq 4 \end{cases}$

Graph
$y > |x + 5| - 3 y \geq - \sqrt{x + 4}$

$y > |x + 5| - 3 absolute value : a = 1, h = - 5, k = - 3 y \geq - \sqrt{x + 4} square root : a = - 1, h = - 4, k = 0 asymptote : x = - 4$

Graph
$y = ⌈x⌉ + 2$

shift up 2 from parent ceiling function

A Wildlife Sanctuary sells three types of day passes: child, student, and adult. The price of one child is $20 (ages 0–10), one student is $25 (ages 11–22), and one adult is $25 (ages 23+).

Write a piecewise function for the types of day passes, $p (d) .$

$p (d) = \{\begin{cases} 20 & if 0 \leq d < 11 \\ 25 & if 11 \leq d < 23 \\ 35 & if d \geq 23 \end{cases}$

Graph the function.

Note

Your student will need to scale both axes. Their scale may look different than the answer key.

Write the domain intervals for the piecewise function.

$domain : \{\begin{cases} - 6 \leq x < 0 \\ 1 \leq x < 5 \end{cases}$

Name the domain and range for the piecewise function.

$domain : \{x | x \in ℝ\} range : \{y | y \in ℤ\}$

A hiker’s GPS tracker recorded the distance the hiker was from the start/end of a loop-trail. The GPS recorded five sections, labeled A–E on the graph.

What is the rate of change for the first 60 minutes of the hike?

$m = \frac{distance}{time} = \frac{1 - 0}{50 - 0} = \frac{1}{50}$

1 mile in 50 minutes

How long did the hike take?

300 minutes or 5 hours

Explain what a slope of zero means in sections B and D of the hike.

Sample: For sections B and D in the hike, the slope of zero means the hiker is resting because time is increasing, but distance remains the same.

What type of slope is section E of the hike? What does this mean in context?

Sample: The slope is negative. The distance is decreasing because as the hiker moves closer to the start of the trail, the time is still increasing.

Write the inequality for the given graph.

$y < \sqrt{x + 4} + 1$