BRIEF ANSWERS FOR TEST 1

1. T or F?
   1. T. Multiplying by 2 can change the range but not the domain.
   2. F. For example, the domain of f(x) = square_root(x-2) is [2,infinity) while the domain of f(2x) is [1,infinity).
   3. T. If y=f(x) is odd, then its graph is symmetric through the origin. The function y=-f(x), whose graph is obtained by reflecting the graph of y=f(x) across the x-axis, is also symmetric through the origin. Hence y=-f(x) is odd too.
   4. F. Width is independent of length.
2. Find the domain ...
   1. (-infinity,-1] union [1,4]
   2. all real numbers except -4.
3. Sketch ...
   • Start with the graph of the absolute value function, stretch vertically by a factor of 2, shift right two units, shift up two units. (If MAPLE is installed on your machine, then you can see the graph here.)
4. Let the dimensions of the box be x, x, and y. Then the volume is V = x^2*y = 12, and the surface area is A = 2*x^2 + 4*x*y. Using the equation x^2*y = 12 we can eliminate x from A = 2*x^2 + 4*x*y. We get:
   A = 2*(12/y) + 4*(square_root(12/y))*y,
   which expresses A as a function of y.
5. We want to maximize the area A = x*y given that x+2*y = 20. Thus we want to maximize A = (20-2*y)*y = -2*y^2 + 20*y = -2*(y-5)^2 + 50. The maximum area occurs when y = 5, and the maximum area is 50 square units.
6. The possible rational roots are +/-1 and +/-2. Testing each one shows that x = 2 and x = -1 are roots. Therefore x^4 + x^3 - 5x^2 - 3x + 2 = (x - 2)(x + 1)(x^2 + 2x - 1). The roots are 2, -1 and the roots of x^2 + 2x - 1. Using the quadratic formula, we get that the full list of roots is 2, -1, -1 + square_root(2), -1 - square_root(2).
7. Use the following information to sketch:
   • x-intercepts: +1 (multiplicity 2), -1 (multiplicity 2), +3 (multiplicity 1), -3 (multiplicity 1).
   • vertical asymptotes: 0 (multiplicity 3), +2 (multiplicity 1), -2 (multiplicity 1).
   • slant asymptote y = x.
   • function is odd.
     (If MAPLE is installed on your machine, then you can see the graph here.)