Mathematics

Mathematics 100 - Precalculus Homework 2

1. Consider the function f(x) = x2 − 2x + 2 with domain [0, 2]. Does f have an inverse? If so, find it. If not, explain why.

2. Find a number t such that the line containing the points (t,−2) and (−3, 5) is perpendicular to the line containing the points (4, 7) and (1, 11).

3. Find the maximum value of the quadratic function 9 + 5x− 4x2.

4. Find the points of intersection between the circle of radius √

2 centered at the origin, and the parabola y = x2.

5. Simplify the expression x−11(y3)−2

(x−3)5(y2)4

6. Find a polynomial p(x) of degree 4 having exactly two zeros: −1 and 1. Hint: Express p(x) as a product of two factors x− t, and a quadratic having no roots.

7. Find all numbers x such that

log3(x) + log3(x + 2) = 1

Note: Since x = logb(y) is equivalent to y = b x where b > 0. This means y must always be

a positive number (any power of a positive number is positive). In other words, logarithms of non-positive numbers are not defined. This fact will be useful for the last question.

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