Basic Properties of Functions

Things I need you to know about functions Things I need you to know about functions. (All of these skills you should be expected to know how to do either with or without a calculator.) How to do basic substitution and recognize points using function notation. How to graph a function. How to find the domain and range of a function. How to find the zeros and y-intercept. How to determine the intervals of increasing and decreasing How to identify asymptotes. How to identify the basic behavior of the function just by looking at the equation

What is a function? The most common name is "f", but you can have other names like "g“ and basically they are all interchangeable with y.

Domain and range Basically the domain is the list of all x values and the range is the list of all y values.

Finding Zeros The zeros of a function f are found by solving the equation f(x) = 0. f(x) = -2 x + 4 -2x + 4 = 0 x = 2

Example - zeros: Find the zeros of the quadratic function f(x) = -2x2-5x+7 Set equal to zero and then Factor! -2x2-5x+7 = 0 (Multiply by -1 first, easier!) 2x2 + 5x - 7 = 0 (2x + 7)(x - 1) = 0 Therefore x = -7/2 and x = 1 

Example - zeros:On TI-89 Find the zeros of the quadratic function f(x) = -2x2-5x+7 2 methods – You need to know both! On home screen – F2 – algebra/solve Your input line should look like this: solve(2x2 + 5x - 7 = 0,x) On graph menu – input function on y= menu go to graph, then F5, find zero key. You will get a prompt for lower bound. Move cursor to left side of zero, then upper bound, mover cursor to right side of zero. If done correctly, it will give you the correct zero. 

Basic Graphs of Functions A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. The graph of a quadratic function is a curve called a parabola.

Parabolas Parabolas have a maximum or minimum point based on whether they point up or down. This point is called the vertex. The formula used to find the x value of the vertex is x = –b/(2a)

Example - Vertex of a Parabola   Example - Vertex of a Parabola Find the vertex of the parabola given by the equation: f(x) = x2-4x+7 -b/2a =-(-4/2(1)), or 2 f(2)=22-4(2)+7, or 3 The vertex of this parabola is (2, 3).

Increasing Functions A function is "increasing" if the y-value increases as the x-value increases, like this

Decreasing Functions A function is decreasing if the y-value decreases as the x-value increases:

Example – Inc/dec The graph of the function f(x) = 1/3x3 + x2 - 8x + 1 is shown here. For what values of x is the function f(x) = 1/3x3 + x2 - 8x + 1 decreasing? On (-4,2) – notice the interval is part of the domain or ‘x’ values!

Basic Graphs of Functions Absolute value of │x│ Domain (-∞,∞) Range: [0,∞)

Basic Graphs of Functions f(x) = √x Domain [0, ∞) Range [0, ∞)

Basic Graphs of Functions f(x) =√1-x2 or (1-x2)1/2 Domain [-1, 1] Range [0, 1]

Asymptotes There are three types of linear asymptotes: horizontal, vertical and oblique:

Asymptotes A graph will approach an asymptote but never touch it.

Example - Asymptotes The function y = 1/x. Horizontal asymptote: y = 0 Vertical asymptote: x = 0 Always write asymptotes as equations of lines

Complete Example -You should be able to successfully find all of this information without a calculator. Find all of the following and graph: f(x) = x2 Classify the function (Is it a polynomial, a trig function, a rational function, etc.) f(4) and f(-4) The vertex, y-intercept, and zeros The domain and range

Example: Answers This function is a monomial (only one term with a positive whole number exponent). Monomials are part of the larger group of functions called polynomials which have multiple terms but still only have positive whole number exponents. It is also a quadratic function or a parabola. f(4)=16; f(-4)=16 or (4,16) and (-4, 16) are 2 points on the graph. The vertex, y-intercept and zero of this function are all at the same point (0,0) The domain is (-∞,∞) and the range is [0,∞) Note: All polynomials have a domain of all Real Numbers Note: Always use parentheses on unbounded sets to infinity Note: Use a bracket on a bounded set if the point is included as in the zero in the range above, a parenthesis if it is not included.

Graph of y = x2