To determine the range of the linear function [tex]\( c(x) = -10x + 150 \)[/tex], which models the number of pieces of candy remaining after a given number of days [tex]\( x \)[/tex], let's analyze the function step-by-step.
### Step 1: Understanding the Variables
- [tex]\( c(x) \)[/tex]: The number of pieces of candy remaining after [tex]\( x \)[/tex] days.
- [tex]\( x \)[/tex]: The number of days passed.
### Step 2: Identifying the Constraints
1. The minimum number of days ([tex]\( x \)[/tex]) is 0 because selling candy does not begin before day 0. Therefore, [tex]\( x \geq 0 \)[/tex].
2. We know each member starts with 150 pieces of candy. So, initially (at [tex]\( x = 0 \)[/tex]):
[tex]\[
c(0) = -10(0) + 150 = 150
\][/tex]
### Step 3: Finding the Specific Points
1. To find the number of days when all the candy is sold, set [tex]\( c(x) = 0 \)[/tex]:
[tex]\[
0 = -10x + 150
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
10x = 150 \implies x = \frac{150}{10} \implies x = 15
\][/tex]
Therefore, when [tex]\( x = 15 \)[/tex]:
[tex]\[
c(15) = -10(15) + 150 = 0
\][/tex]
### Step 4: Determine the Range
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 15 \)[/tex]:
- At [tex]\( x = 0 \)[/tex], [tex]\( c(x) = 150 \)[/tex].
- At [tex]\( x = 15 \)[/tex], [tex]\( c(x) = 0 \)[/tex].
Thus, as [tex]\( x \)[/tex] goes from [tex]\( 0 \)[/tex] to [tex]\( 15 \)[/tex], [tex]\( c(x) \)[/tex] decreases linearly from [tex]\( 150 \)[/tex] to [tex]\( 0 \)[/tex].
Therefore, the range of the function [tex]\( c(x) = -10x + 150 \)[/tex] is:
[tex]\[
[0, 150]
\][/tex]
So, the correct range of the function is:
[tex]\[ \boxed{[0, 150]} \][/tex]
