170- Quiz 1

Quiz 1 is here.

Solutions follow.

Problem 1 asks to find the domain of the function \displaystyle y=\frac1{-1+\sqrt x}.

For this, note that we need to satisfy two requirements:

  • The denominator -1+\sqrt x cannot be 0. This is the same as asking that x\ne1.
  • \sqrt x must be defined, so x\ge0.

Combining these two requirements, we see that the domain consists of those values of x that are non-negative, and different form 1. We can write this as 0\le x, x\ne1, or as {}[0,1)\cup(1,\infty).

Problem 2 is a word problem: A car rental firm has the following charges for a certain car: $25 per day with 100 free miles included, $0.15 per mile for more than 100 miles. Assuming the car is only rented for one day, draw a graph relating the cost y to the number of miles x that the car is driven. Write an equation for y as a function of x.

(The requirement that the car is only rented for one day was left out from the original wording, sorry for any confusion this may have caused.)

The graph of the function y is displayed above (click to enlarge). Note that the function behaves differently depending on whether x\le 100 (in which case, the only cost is the $25 charged up front) or x\ge 100 (in which case, in addition, we are charged per mile.)

The equation for y when x\le 100 is simply y=25.

The equation for y when x\ge 100 has the form y=mx+b. We have that m=0.15, since we are charged additional 15 cents per mile. To find b, we use that y=25 when x=100, so 25=0.15(100)+b, or b=25-15=10. Finally, we see that y=0.15x+10. Again, this is only valid for x\ge 100.


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