Let's determine the domain and range of the function [tex]\( V = f(t) = 24000 - 3400t \)[/tex], which represents the price [tex]\( V \)[/tex] of a car that is [tex]\( t \)[/tex] years old.
### Domain
To find the domain of the function [tex]\( f(t) \)[/tex], we need to identify all possible values of [tex]\( t \)[/tex]:
- Since [tex]\( t \)[/tex] represents the number of years a car has been used, [tex]\( t \)[/tex] must be a non-negative number. Therefore, [tex]\( t \geq 0 \)[/tex].
- To find the upper bound for [tex]\( t \)[/tex], we need to determine when the car's value [tex]\( V \)[/tex] would essentially become zero:
[tex]\[
V = 24000 - 3400t
\][/tex]
Setting [tex]\( V \)[/tex] to 0 and solving for [tex]\( t \)[/tex]:
[tex]\[
0 = 24000 - 3400t
\][/tex]
[tex]\[
3400t = 24000
\][/tex]
[tex]\[
t = \frac{24000}{3400} = 7.0588235294117645
\][/tex]
Therefore, the domain of [tex]\( f(t) \)[/tex] is the interval [tex]\( [0, 7.0588235294117645] \)[/tex].
Domain: [tex]\( 0 \leq t \leq 7.0588235294117645 \)[/tex]
### Range
To find the range of the function [tex]\( f(t) \)[/tex], we need to identify all possible values of [tex]\( V \)[/tex]:
- When [tex]\( t \)[/tex] is at its minimum value (0 years), [tex]\( V \)[/tex] is at its maximum value:
[tex]\[
f(0) = 24000 - 3400 \cdot 0 = 24000
\][/tex]
- When [tex]\( t \)[/tex] is at its maximum value (approximately 7.0588235294117645 years), [tex]\( V \)[/tex] is at its minimum value:
[tex]\[
f \left( 7.0588235294117645 \right) = 24000 - 3400 \cdot 7.0588235294117645 = 0
\][/tex]
Therefore, the range of [tex]\( f(t) \)[/tex] is the interval [tex]\( [0, 24000] \)[/tex].
Range: [tex]\( 0 \leq V \leq 24000 \)[/tex]
Putting it all together:
Domain: [tex]\( 0 \leq t \leq 7.0588235294117645 \)[/tex]
Range: [tex]\( 0 \leq V \leq 24000 \)[/tex]
