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Section 1-2

Section 1-2. Functions and Their Properties. Section 1-2. function definition and notation domain and range continuity increasing/decreasing boundedness local and absolute extrema symmetry asymptotes end behavior. Functions.

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Section 1-2

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  1. Section 1-2 Functions and Their Properties

  2. Section 1-2 • function definition and notation • domain and range • continuity • increasing/decreasing • boundedness • local and absolute extrema • symmetry • asymptotes • end behavior

  3. 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-value • the set of x-values (independent variables) is called the domain • the set of y-values (dependent variables) is called the range

  4. Function Notation • when working with functions, equations like y = x2 are written as f (x) = x2 • 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

  5. 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

  6. Domain and Range • the range is all the resulting y-values • 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

  7. 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

  8. 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

  9. 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

  10. 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)

  11. Symmetry • graphs that look the same on the left side of the y-axis as the right side are said to have y-axis symmetry • if a graph has y-axis symmetry then the function is even • the algebraic test for y-axis sym. is to plug –x into the function, and it can be simplified back into the original function

  12. Symmetry • graphs that look the same above the x-axis as below are said to have x-axis symmetry • the algebraic test for x-axis sym. is to plug –y into the equation, and it can be simplified back into the original equation • graphs with x-axis symmetry are usually not a function

  13. 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

  14. 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

  15. Limit Notation • a new type of notation used to describe the behavior of curves near asymptotes (and for many other things as well) • represents the value that the function f (x) approaches as x gets closer to a

  16. Asymptote Notation • if y = b is a horizontal asymptote of f (x), then • if x = a is a vertical asymptote of f (x), then

  17. End Behavior • end behavior is a description of the nature of the curve for very large and small x-values (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) = x2 then

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