Let's analyze the linear function [tex]\( c(x) = -10x + 140 \)[/tex] which models the number of pieces of candy remaining after [tex]\( x \)[/tex] days.
1. Interpret the function:
- The function [tex]\( c(x) = -10x + 140 \)[/tex] starts with 140 pieces of candy (when [tex]\( x = 0 \)[/tex]).
- As each day passes (increasing [tex]\( x \)[/tex]), the number of remaining pieces decreases by 10 pieces per day (since the coefficient of [tex]\( x \)[/tex] is -10).
2. Find the range of [tex]\( c(x) \)[/tex]:
- When [tex]\( x = 0 \)[/tex], the function value is:
[tex]\( c(0) = -10(0) + 140 = 140 \)[/tex]
- For any positive value of [tex]\( x \)[/tex]:
[tex]\( x > 0 \)[/tex], the term [tex]\(-10x\)[/tex] will be subtracted from 140, which means the function value will decrease.
3. Determine the lowest possible value of [tex]\( c(x) \)[/tex]:
- [tex]\( c(x) \)[/tex] will continue to decrease until it reaches zero. To find the point at which [tex]\( c(x) = 0 \)[/tex]:
[tex]\[
-10x + 140 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
-10x = -140 \\
x = 14
\][/tex]
- Therefore, after 14 days, the number of pieces of candy will be zero.
4. Confirm the range:
- Before day 14, [tex]\( c(x) \)[/tex] will be a value between 0 and 140.
- After 14 days, the number of pieces of candy cannot become negative, meaning the lowest value [tex]\( c(x) \)[/tex] can take is 0.
5. Conclusion:
The range of the function [tex]\( c(x) \)[/tex] is from 0 to 140, inclusive.
Therefore, the correct answer is:
[tex]\[ [0, 140] \][/tex]
