Section Functions. Function Notation. Is this a function?

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1 Section 1-21 Functions and Their Properties Section 1-21 function definition and notation domain and range continuity increasing/decreasing boundedness local and absolute extrema symmetry asymptotes end behavior Functions a relation is a set of ordered pairs (list, table, graph, mapping) a function is a relation in which each x-value corresponds to exactly one y-valuey the set of x-values x (independent variables) is called the domain the set of y-values y (dependent variables) is called the range Function Notation when working with functions, equations like y = x 2 are written as f (x)) = x 2 f (3) means to input 3 into the function for x to find the corresponding output ordered pairs then become (x(, f (x)) ) instead of (x(, y) for the graph of a relation to be a function, it must pass the vertical line test Is this a function? Domain and Range the domain of a function is all of the possible x-values usually the domain is all reals,, but values that make the denominator = 0 or a square root of a negative must be excluded other excluded values will arise as we study more complicated functions the domain of a model is only the values that fit the situation 1

2 f ( x) = x+ 3 x g( x) = x Find the domain 2 ( ) = s where A(s) is the area of equilateral triangle. A s Domain and Range the range is all the resulting y-valuesy the easiest way to find the range is to examine the graph of the function if you cannot graph it, then try plugging in different types of domain values to see what types of outputs are created Find the range 2 f ( x) x = 2 10 x = g( x) Continuity a graph is continuous if it can be sketched without lifting your pencil graphs become discontinuous 3 ways holes (removable( discontinuity) jumps (jump discontinuity) vertical asymptotes (infinite discontinuity) continuity can relate to the whole function, an interval, or a particular point Tell if the graph of the function has a discontinuity at x = a and what type Increasing/Decreasing at a particular point, a function can be increasing, decreasing, or constant it is easiest to figure out by looking at a graph of the function if the graph has positive slope at the point then it is increasing, negative slope then it is decreasing, and slope=0 then it is constant we use interval notation to describe where a function is increasing, decreasing, or constant 2

3 Tell the intervals on which the function is increasing, decreasing or constant Boundedness if a graph has a minimum value then we say that it is bounded below if a graph has a maximum value then we say that it is bounded above a graph that has neither a min or max is said to be unbounded a graph that has both a min and max is said to be bounded Tell whether the function is bounded above, bound below, bounded or unbounded Local and Absolute Extrema if a graph changes from increasing to decreasing and vice versa, then it has peaks and valleys the point at the tip of a peak is called a local max the point at the bottom of a valley is called a local min if the point is the maximum value of the whole function then it can also be called an absolute max (similar for absolute min) Find all the local max and min and where they occur on grapher Symmetry (day #2) graphs that look the same on the left side of the y-axis y as the right side are said to have y-axis symmetry if a graph has y-axis y symmetry then the function is even the algebraic test for y-axis y sym. is to plug x into the function, and it can be simplified back into the original function f ( x) = f ( x) 3

4 Symmetry graphs that look the same above the x-axis x as below are said to have x-axis symmetry the algebraic test for x-axis x sym. is to plug y into the equation, and it can be simplified back into the original equation graphs with x-axis x symmetry are usually not a function Symmetry graphs that can be rotated 180 and still look the same are said to have symmetry with respect to the origin if a graph has origin symmetry then the function is called odd the algebraic test for origin symmetry is to plug x into the function, and it can be simplified back into the opposite of the original function f ( x ) = f ( x ) Asymptotes some curves appear to flatten out to the right and to the left, the imaginary line they approach is called a horizontal asymptote some curves appear to approach an imaginary vertical line as they approach a specific value (they have a steep climb or dive), these are called vertical asymptotes we use dashed lines for an asymptote 4

5 How to find the asymptotes Vertical asymptotes are found by setting the denominator equal to zero and solving Horizontal asymptotes are found by using the degree of the functions If higher on top, then no horiz asym. If higher on bottom, then y = 0 is horiz asym. If equal, then the horiz asym is y=a/b where a and b are the leading coefficients Limit Notation a new type of notation used to describe the behavior of curves near asymptotes (and for many other things as well) lim f ( x ) x a represents the value that the function f (x)) approaches as x gets closer to a Asymptote Notation if y = b is a horizontal asymptote of f (x), then lim f ( x) = b or lim f ( x) = b x x if x = a is a vertical asymptote of f (x), then lim f ( x) =± or lim f ( x) =± _ x a x a+ End Behavior end behavior is a description of the nature of the curve for very large and small x-values x (as x approaches ± ) a horizontal asymptote gives one type of end behavior if the f(x) ) values continue to increase (or decrease) as the x values approach ± then a limit can be used to describe the behavior ex. If f (x)) = x 2 then lim f ( x) = x ± 5

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