Sam Miner Department of Mathematics Dartmouth College Math 3, Fall 2015 Sam Miner Department of Mathematics Dartmouth College Class 2 - 9/18/2015

A domain problem Let f(x) = √x, and g(x) = 4 – x^2. What is the domain of f °g(x)? {x | |x| ≥ 2 } {x | |x| ≤ 2} {x | x > 2} {x | x < 2} As 4-x^2 and is not defined for we have that is not defined for 4-x^2. This reduces to x^2 or . The domain is thus all other values. Class 2 - 9/18/2015

Even and odd Suppose f is odd and g is even. Then fg is Odd Even Odd and even Neither What are the definitions? Can a function be both even and odd? Neither even nor odd? Class 2 - 9/18/2015

Nomenclature Let f(x) = (x^2 + 2x – 3) / (1-√x). Then f is a polynomial a rational function a trigonometric function an algebraic function What do each of these terms mean? Why the distinction between rational and algebraic? Class 2 - 9/18/2015

Even, Odd, Both, or Neither? Paired exercise: 2 students one solver, one listener. One problem each switching roles. The solver talks through and explains the solution to the problem. The listener follows the exposition and asks questions to clarify the answer as best they can. The listener should prompt but not help (yet) – call on me for help if you get stuck. BE KIND! Use phrases like I don’t understand X Why is Y true? I think we need to check Z. Class 2 - 9/18/2015

Even and odd functions Is f(x) = 2x^5 – x^3 even, odd, both, or neither? Same pair – do this problem together. Make sure either one can present a solution. Class 2 - 9/18/2015

Domain and range Randomize pairs (find a new partner you don’t know). Same technique – if you went first last time, go second (if both 1st or 2nd, flip coin). Find the domain and range of the two functions pictured above. Class 2 - 9/18/2015

Domain and range Find the domain and range of the function f(x) = 3 - √(2-x). Consider the function g(x) = 4/(3-x). Are there any values which are not in the domain of g? What about the range? Same pair – do this problem together. Make sure either one can present a solution. Class 2 - 9/18/2015