1.2 Functions & their properties

Notes 9/28 or 10/1

Relations

• Every relation has a domain and range• Domain : x values, independent• Range: y values, dependent • Functions: x value DO NOT REPEATExamples: {(12, 4), (8, 3), (3, 9)}

domain: {3, 8, 12}, range: {3, 4, 9}, is a function

• From a graph:

• Find domain or range when given an equation:• -determine what values of x will work• Ex: 1) f(x) = x + 4

• 2) f(x) = x – 10

• 3) f(x) = 5 x - 5

Continuity

• Continuous: continuous at all values of x• Discontinuity: examples on p. 84-85

- removable discontinuity: there is a “hole” in your graph- jump discontinuity: the graph “jumps” a point(s)- infinite discontinuity: the graph has a vertical asymptote (there is a vertical line where the graph cannot cross or touch)

Identify pts of discontinuity

• Graph it, also see when the denominator = 0Ex: 1)f(x) = x + 3 x – 2

2)f(x) = x2 + x – 6

3) f(x) = x2 – 4 x - 2

Increasing/Decreasing function

• Functions can be increasing, decreasing, or constant• A function is increasing on an interval if, for any 2

pts in the interval, a positive change in x results in a positive change in f(x)

• A function is decreasing on an interval if, for any 2 pts in the interval, a positive change in x results in a negative change in f(x)

• A function is constant on an interval if, for any 2 pts in the interval, a positive change in x results in a 0 change in f(x)

• Determining increasing/decreasing intervals:look for the x values that the graph is increasing/decreasing/constant

• #1 and 2 on handout• 3) f(x) = 3x2 - 4

Boundedness

• A function is bounded below if there is a minimum. Any such # b is called a lower bound of the function.

• A function is bounded above if there is a maximum. Any such # B is called an upper bound of the function.

• A function f is bounded if it is bounded both above and below

Examples of bounded

• Bounded below:

• Bounded above

• bounded

Local & Absolute Exterma

• Maximums/minimums – every function (w/ the exemption of a linear function)

• To determine where the local maximum and/or local minimum is located look at the graph or use a calculator

• Ex: #8 & 9 on handout • 3) f(x) = -x2 – 4x + 5• 4) f(x) = x3 – 2x + 6

Asymptotes

• Vertical asymptotes (VA): set the denominator = 0 and solve, write answers as equations of vertical lines (x = #)

• Horizontal asymptotes (HA): 3 possibilities 1) if the exponent is lower in the numerator then the

denominator: the HA is y = 02) if the exponents are equal: the HA is y = a/b, where a is the leading coefficient in the numerator & b is the leading coefficient in the denominator3) if the exponent is higher in the numerator than the denominator there is no HA

examples

• Identify the asymptotes:• 1) f(x) = 3x x2 - 42) f(x)= 3x2

x2 – x - 23) f(x)= 3x3

x + 2x2 - 15

End behavior

• What direction does the graph go (up or down) at the far left and far right

Ex: 1) f(x)= 3x x2 - 12) f(x)= 3x2

x2 - 13) f(x)= 3x3

x2 - 1

Homework

• Section 1.2 exercises p. 94-95 #2-16 even, 17-28 all, 36-46 even, 56-62 even