Intro to FunctionsMr. GonzalezAlgebra 2

Linear Function (Odd)• Domain(-, )• Range(- , )• Increasing (- , )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)-

Quadratic Function (Even)• Domain(-, )• Range[0, )• Increasing (0, )• Decreasing(-, 0)• End BehaviorAs x, f(x)As x-, f(x)

Cubic Function (Odd)• Domain(-, )• Range(- , )• Increasing (- , 0)(0, )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)-

Absolute Value Function (Even)• Domain(-, )• Range[0, )• Increasing (0, )• Decreasing(- , 0)• End BehaviorAs x, f(x)As x-, f(x)

Square Root Function (Neither)• Domain[0, )• Range[0, )• Increasing (0, )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)0

Cube Root Function (Odd)• Domain(-, )• Range(-, )• Increasing (-,0)(0, )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)-

Exponential Function (Neither)• Domain(-, )• Range(0, )• Increasing (-, )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)0

Logarithmic Function (Neither)• Domain(0, )• Range(-, )• Increasing (-, )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)-

Inverse Function (Odd)• Domain(-, 0)(0, )• Range(-, 0)(0, )• Increasing Never• Decreasing(-,0)(0, )• End BehaviorAs x, f(x)0As x-, f(x)0

Inverse Squared Function (Even)

• Domain(-, 0)(0, )• Range(0, )• Increasing (-,0)• Decreasing(0, )• End BehaviorAs x, f(x)0As x-, f(x)0

Constant Functions (Even/Neither)

Horizontal• Domain(-, )• Range(y)

Vertical• Domain(x)• Range(-, )

Step Function (Neither)• Domain(-, )• Range(only integers)• Increasing Never• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)-

Maximum and Minimums (Extrema) Absolute Max and Min Relative Max and Min

• We will have an absolute maximum (or minimum) at  provided f(x) is the largest (or smallest) value that the function will ever take on the domain that we are working on. There may be other values of x that we can actually plug into the function but have excluded them for some reason.

• A relative maximum or minimum is slightly different.  All that’s required for a point to be a relative maximum or minimum is for that point to be a maximum or minimum in some interval of x’s around .  There may be larger or smaller values of the function at some other place, but relative to, or local to ,  f(c) is larger or smaller than all the other function values that are near it.

Example #1• Domain(-, )• Range(-, )• Increasing (- ,-2)(0, )• Decreasing(-2, 0)• End BehaviorAs x, f(x)As x-, f(x)-• ExtremaRelative Max at y=3Relative Min at y=-3

Example #2• Domain(-, )• Range[-7, )• Increasing (-0.5, 1.5)(2.5, )• Decreasing(- , -0.5)(1.5, 2.5)• End BehaviorAs x, f(x)As x-, f(x)• ExtremaAbsolute Min at y=-7Relative Max at y=1Relative Min at y=-1.5

Example #3• Domain(-, )• Range(-, 17]• Increasing (-, -0.5)(0.5, 1.5) (2, 2.5)• Decreasing(-0.5, 0.5) (1.5, 2) (2.5, )• End BehaviorAs x, f(x)-As x-, f(x)-• ExtremaAbsolute Max at y=17Relative Max at y=0.5 and y=1Relative Min at y=-3.5 and y=0

Example #4• Domain[-5, )• Range[-6, )• Increasing [-5, )• DecreasingNever• End BehaviorAs x, f(x)As x-, f(x)-6• ExtremaAbsolute Min at y=-6

Example #5 (Piecewise Functions)

• Domain(-, -1)[-1, ) or (-, )• Range[-3, )• Increasing (0, )• Decreasing[-1, 0)• Constant(-, -1)• End BehaviorAs x, f(x)As x-, f(x)1• ExtremaAbsolute Min at y=-3Relative Max at y=-2

Example #6 (Piecewise Functions)

• Domain(- 1](1, 2)[2, ) or (-, )• Range(-, )• Increasing (-, 1)(2, )• Decreasing(1, 2)• End BehaviorAs x, f(x)As x-, f(x)-• ExtremaRelative Max at y=4Relative Min at y=-2

Example #7• Domain(-, )• Range[0, )• Increasing (-1, 0.5)(2, )• Decreasing(- , -1)(0.5, 2)• End BehaviorAs x, f(x)As x-, f(x)• ExtremaAbsolute Min at y=0Relative Max at y=2

Example #8 (Piecewise Functions)

• Domain(-, 2](2, ) or (-, )• Range[0, )• Increasing (0, 2]• Decreasing[-, 0)• Constant(2, )• End BehaviorAs x, f(x)3As x-, f(x)• ExtremaAbsolute Min at y=0Relative Max at y=2