Question

michelle and rashid love going on long bike rides. every saturday, they have a particular route they bike together that takes four hours. given is a piece - wise function that estimates the distance they travel in kilometers for each hour of their bike ride.
f(x)=\begin{cases}16x, & 0leq xleq1\\10(x - 1)+16, & 1lt xleq2\\14(x - 2)+26, & 2lt xleq3\\12(x - 3)+40, & 3lt xleq4end{cases}

1. what part of the bike ride are they going the fastest? slowest?
2. what is the domain of (f(x))?
3. find (f(2)). explain what this means in terms of the context.
4. how far have they traveled at 3 hours? write the answer using function notation.
5. what is the total distance they travel on this bike ride?

Explanation:

Step1: Identify the function for $x = 2$

Since $1<2\leq2$, we use $f(x)=10(x - 1)+16$.

Step2: Substitute $x = 2$ into the function

$f(2)=10\times(2 - 1)+16=10\times1 + 16=26$. In the context, it means that after 2 hours of the bike - ride, Michelle and Rashid have traveled 26 kilometers.

Step3: Find the distance at 3 hours

Since $2<3\leq3$, we use $f(x)=14(x - 2)+26$. Substitute $x = 3$: $f(3)=14\times(3 - 2)+26=14\times1+26 = 40$. So they have traveled 40 kilometers at 3 hours.

Step4: Find the total distance (at $x = 4$)

Since $3<4\leq4$, we use $f(x)=12(x - 3)+40$. Substitute $x = 4$: $f(4)=12\times(4 - 3)+40=12\times1+40=52$ kilometers.

Answer:

1. $f(2) = 26$, which means they have traveled 26 km after 2 hours.
2. $f(3)=40$, they have traveled 40 km at 3 hours.
3. 52 km, which is the total distance they travel on the 4 - hour bike ride.