Functions (Sections 1.2 & 1.3)

Transcription

1 Functions (Sections. &.) A function is a rule (or set of instructions or relation) that assigns each value in one set (called the domain) to EXACTLY ONE value in another set (called the range). The independent variable (or input) represents arbitrary values in the domain. The dependent variable (or output) represents arbitrary values in the range. Function notation. We name functions (for example, the squaring function, the absolute value function, and the cube root function) or use letters to represent them (usually, but not limited to, f, g, or h). The output of the function f when x is the input is written f x which is read "f of x" and is called the value of f at x. For example, the squaring function can be written as f x x. In this case, the function f has the rule that every input x is squared (x ) to get the corresponding output. Sometimes the f x notation is not used as in the function y x. Here y is the dependent variable or output and x is the independent variable or input. In this case we say y is a function of x. Here are some more examples of functions:. A r r ; This is the function that gives the area, A, of a circle as a function of its radius, r. The independent variable is r.. g u u ; The function g has a rule that cubes any input u.. y 6t 0t; We have y as a function of t. The dependent variable is y and the independent variable is t. Evaluating functions. If g x x x we think of x as just a place holder or "box": g. Thus we have g which indicates that if we input into this function we get as the corresponding (or assigned) output: g. Here are some more evaluations for this function: g 0 g

2 Sometimes the input is a variable or an expression involving the variable. We evaluate as above and then try to simplify the output if we can. Here are some examples using the same function g x x x above: If g x x x, Evaluate. g t g t g a a a g s s s s s g a h a h a h a ah h a h a 6ah h a h. g a h g a h

3 The graph of the function f is the set of ordered pairs x, f x where x is in the domain of the function f. We can visualize the graph by plotting the ordered pairs of the graph in the coordinate plane (we often call this visualization the graph of the function). Use the graph of the function f shown below to estimate the following: f 0 f f If f x 0, what are possible values for x? Find the value of f f. What does this value represent graphically? Vertical Line Test Recall a function assigns each input (independent variable) to exactly one output (dependent variable). Thus, a curve in the coordinate plane is a function if and only if all vertical lines intersect the curve at most once. Thus a curve is NOT a function if you can draw at least one vertical line that intersects the curve more than once.

4 Consider the graph of the functions shown below. What are the domains and ranges of these functions? Domain of h : Range of h : Graph of g Domain of f : Range of f : -6 Graph of h Domain of h : Range of h : - Graph of h

5 Increasing, Decreasing, & Constant Functions. A function is increasing on an interval if f a f b whenever a b. A function is decreasing on an interval if f a f b whenever a b. A function is constant on an interval if f a f b for all values a and b in the interval. Increasing Decreasing Constant Over what intervals is the function shown below increasing, decreasing, and constant? The function is increasing on The function is decreasing on The function is constant on

6 Relative Maxima & Relative Minima A function f has a relative maximum of f a that occurs at x a if f a f x for all x in an open interval about a. A function f has a relative minimum of f a that occurs at x a if f a f x for all x in an open interval about a. Relative maxima & minima are sometimes called relative extrema (singular is extremum). Find all relative maxima & minima for the function f whose graph is shown below f has a relative maximum of that occurs at x f has a relative minimum of that occurs at x 6

7 Even & Odd Functions and Symmetry. A function f is called an even function if f x f x for all x in the domain of f. Even functions have what we call y-axis symmetry. Example: f x x A function f is called an odd function if f x f x for all x in the domain of f. Alternatively, we can say f is an odd function if f x f x. Odd functions have what we call origin symmetry. Example: f x x Determine whether the function is even, odd, or neither: g x x h x x 7 7

8 Sketching the graph of a function Sketch the graphs of the following functions by plotting points. x 0 g x g x x 0 h x h x x t 0 y y t 8

9 You must know the graphs of the following basic functions. Know their domains, ranges, and general shape. Constant functions: f x c, where c is any constant Linear functions: f x mx b, where m and b are constants Squaring function: f x x 0-0 Cubing function: f x x Square root function: f x x Cube root function: f x x Absolute value function: f x x Reciprocal function: f x x

10 Piecewise defined functions. Some functions have different rules for different intervals of the domain. Such functions are called piecewise defined functions. Here s an example: h x x x if x x 7 if x Given this function, evaluate: h, h, and h 0. A well-know piecewise defined function is the absolute value function: Calculator notation x abs x x if x 0 x if x 0 0

11 The difference quotient A secant line for a function is a line connecting any two distinct points on the graph of the function. Consider the graph of the function f below. Draw a secant line and find its slope. Evaluate & simplify the difference quotient. f x x x f x h f x h for the following functions:

12 . f x x

13 . f x x

14 Sketch the graph of a function whose domain is all real numbers and that satisfies the given requirements. There are many possible graphs that can be drawn. f, f, f 0, f is increasing on the intervals, and 0, f is decreasing on the interval,