4.+Functions+-+Misconceptions

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Piecewise Functions

 * Some students will find piecewise functions confusing, however they make an excellent object to help solidify the concept of a function.
 * It may be best to look at pictures of piecewise function graphs //prior// to learning about function notation.
 * A good starting place : []

Function Notation
math f(x) = 3x +1, g(w) = w^2. math Evaluate math $f(g(2)), g(f(2))$ math and write an expression for math g(f(z)) math
 * Working with composite functions are an excellent way to clarify and practice function notation.
 * The function notation f(x) looks awfully like f times (x) to many students. Make it explicit the symbol f is a single unit - with the 'x' value put into the function, to generate the value f(x).

Odd and Even Functions

 * **Why do we care about odd and even functions?** One practical reason: it helps with curve sketching.
 * A function may be neither odd //nor// even. In fact, most functions are neither. Some students hear the words odd and even and make the analogy to integers, were every integer fits into one of these two categories.

So math x^2 + 3^x math is not an even function.
 * A function with an axis of symmetry isn't automatically even - the axis of symmetry has to be the y-axis.
 * **Extension work:** Explore how any function can be rewritten as the sum of an odd and even function. See [] (section: Algebraic Structure) if you haven't seen this before.