Testing for even and odd functions

When the end points are included [ ]. When the end points are not included ( ). (4,8) Domain from (2, -3) to (5, -1) Written as [2, 5) Range [ -3, 8] open and close becomes a big deal (2, -3)(5,-1)

Graphically using the Vertical line test. “ A set of points in a coordinate plane is the graph of y as a function of x iff no vertical line intersect the graph at more than one point.”Not a Function Function

Zeros are the x’s that make f(x) = 0 Find the zero of the function f(x) = x 3 -4x 2 + 2x - 8 How do you find them?

Zeros are the x’s that make f(x) = 0 Find the zero of the function f(x) = x 3 - 4x 2 + 2x - 8 How do you find them? Factoring would work

f(x) = x 3 -4x 2 + 2x – 8 f(x) = x 2 (x - 4) + 2(x - 4)

f(x) = x 3 -4x 2 + 2x – 8 f(x) = x 2 (x - 4) + 2(x - 4) f(x) = (x – 4)(x 2 + 2) 0 = (x – 4) and 0 = (x 2 + 2), 4 = x- 2 = x 2 thus the only real answer is x = 4

We only worry about the numerator. 0 = 2a – 6 a = 3

“Increasing” function x 1 f (x 1 ) “Decreasing” functionx 3 f (x 4 ) f(2) f(3) x 1 x 2 x 3 x 4

Here f(2) f(3) x 1 x 2 x 3 x 4

Over a Given Interval Minimum is the lowest point Maximum is the highest point. This will lead to the “Extreme Value Theorem”

EVEN function is where f(x) = f(- x) Odd function is where f(- x) = - f(x) Let g(x) = x 3 + x thus ( -x) 3 + (- x) so - x 3 – x ; - g(x) = - (x 3 + x) It is then Odd f(x) = x thus f(-x) = (-x) ; x which is the same as f(x)It is then Even

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