Quiz 3 (Solution)

Date: MATH 1090-4 - Fall 2003

1. The graph of $f(x)=2x-x^2$ is shown in Figure 1.

   Figure 1: Graph of $f$ for exercise 1

   
   1. What are the coordinates of the point $Q$?

      From the figure we see that they are $(1,1)$.

   2. What are the $x$-coordinates of the points on the graph whose $y$-coordinates are 0?

      From the figure we see that the only two points with $y$-coordinate zero are $(0,0)$ and $(2,0)$. So, the $x$-coordinates in question are 0 and $2$.

   3. If the coordinates of the point $P$ on the graph are $(a,b)$, how are $a$ and $b$ related?

      Since $P = (a,b)$ is on the graph it has to satisfy the equation $y = f(x) = 2x-x^2$. So, $a$ and $b$ are related by $b = 2a-a^2$.

2. Let $A(x) = x^2$ and $B(x) = 2x-1$. Find:
   1. $(A-B)(x)$.

      $$(A-B)(x) = x^2 - (2x-1) = x^2 -2x+1.$$

      
      

   2. $(A\circ B)(x)$.

      $$(A\circ B)(x) = A(B(x)) = A(2x-1) = (2x-1)^2 = 4x^2-4x+1.$$

      
      

   3. $(B\circ A)(x)$.

      $$(B\circ A)(x) = B(A(x)) = B(x^2) = 2x^2-1.$$

      
      

3. What is the domain of $f(x) = \sqrt{x+1}$?

   Since the square root is well defined as long as the radicand (the number whose root you want to find) is $\geq 0$, the domain is $[-1,\infty)$ because adding $1$ to any of these numbers makes it $\geq 0$.

4. A museum's entrance ticket costs $ 10 for each person. They also offer a discounted price for groups of, at least, 9 people. For each person above the number 8, the price of each ticket is reduced by 10 cents.

   For instance, for 9 people each ticket costs $ 9.9 and the total price is $9\times 9.9 = 89.1$.

   If $n$ is the number of people in excess of 8 in a group, express the total price for the group as a function of $n$ (assume $n\geq 0$).

   We know that the total price is $P = ($unit cost of ticket $) \times ($number of people in the group$)$. If we assume that $n\geq 0$ (that is, the total number of people is, at least, $8$) we have that the total number of people is $8+n$ and, because of the discount scheme, the cost of each ticket is $10 - 0.1\cdot n$.

   All together, the price is given by the function

   $$P(n) = (10 - 0.1\cdot n) \cdot (8+n) = 80 + 10n -0.8n -0.1n^2 = 80 + 9.2n -0.1 n^2.$$