1.2 Functions & their properties

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 REPEAT Examples: {(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 = 0 Ex: 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 = 0 2) 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 denominator 3) if the exponent is higher in the numerator than the denominator there is no HA

examples • Identify the asymptotes: • 1) f(x) = 3x x2 - 4 2) f(x)= 3x2 x2 – x - 2 3) 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- 1 2) f(x)= 3x2 x2- 1 3) 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

1.2 Functions & their properties