Part 5: Even and Odd Functions

Even and Odd Functions An even function has a graph which is symmetric with respect to the y –axis: whenever (x, y) is on the graph, then so is the point (–x, y). An odd function has a graph which is symmetric with respect to the origin: whenever (x, y) is on the graph, then so is the point (–x, –y). Example: 𝑦= 𝑥 2 Example: 𝑓 𝑥 = 𝑥 3

Even and Odd Functions A graph that is symmetric with respect to the x-axis is not the graph of a function (except for the graph of y = 0). Symmetric to y-axis Even function Symmetric to origin Odd function Symmetric to x-axis Not a function

Test for Even and Odd Functions… EVEN FUNCTION ODD FUNCTION 𝒇 −𝒙 =𝒇(𝒙) If you plug in a –x into the equation, then you get back the original equation, unchanged. 𝒇 −𝒙 =−𝒇(𝒙) If you plug in a –x into the equation then you get back the negative of the original equation. Example: 𝑓 𝑥 = 𝑥 2 𝑓 −𝑥 = −𝑥 2 = 𝑥 2 Example: 𝑓 𝑥 = 𝑥 3 𝑓 −𝑥 = −𝑥 3 = −𝑥 3

Example 10 – Even and Odd Functions Determine whether each function is even, odd, or neither. g(x) = x3 – x b. h(x) = x2 + 1 c. f (x) = x3 – 1 d. k(x) = 2x + 3

Classwork…