ALGEBRA II HONORS/GIFTED - INVERSES

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Presentation transcript:

ALGEBRA II HONORS/GIFTED - INVERSES 12/7/2018 ALGEBRA II HONORS/GIFTED @ INVERSE RELATIONS and FUNCTIONS

We have a computer called the Gizmoraptor that does computations for us. Q 27 5 -13 -3 R=5Q + 2 3 2 11) What are the ordered pairs we just made? (5, 27), (-3, -13), ( , 3), and (0, 2)

Now, just for fun, let’s make the Gizmoraptor go backwards Now, just for fun, let’s make the Gizmoraptor go backwards. Of course, the equation must change. R Q 27 5 -13 -3 3 2 11) What are the ordered pairs we just made? (27, 5), (-13, -3), (3, ), and (2, 0) 12) What do you notice about the ordered pairs from the two functions R = 5Q + 2 and ?

*Get a sheet of graph paper and draw a Cartesian Coordinate System. *Draw a figure in the 2nd quadrant and note a few of its ordered pairs. *Draw the line y = x on your coordinate system. We note this line has special properties. *Fold your coordinate system using the line y = x as the seam of the fold. *Trace your figure, then draw it on your coordinate system. *What do you notice about the ordered pairs of the second figure you drew?

Recall the equations from before : R = 5Q + 2 which becomes f(x) = 5x + 2 and which becomes 1)a) Find f(g(2)) b) Find g(f(2)) c) Find f(g(-3)) d) Find g(f(-3)) e) Find f(g(x)) f) Find g(f(x))

PROPERTIES OF INVERSES The ordered pairs are reversed. The graphs are reflected about the line y = x. 3) f(g(x)) = g(f(x)) = x For additional information, access http://www.purplemath.com/modules/invrsfcn.htm

ALGEBRA II HONORS/GIFTED - INVERSES 12/7/2018 HORIZONTAL LINE TEST : The inverse of a function, f, is also a function if and only if no horizontal line intersects the graph of f no more than once. EXAMPLES ONE TO ONE FUNCTION : passes both the horizontal and vertical line tests.

Find the inverse. f(x) = 5x + 2 original problem y = 5x + 2 let y be f(x) x = 5y + 2 switch the x’s and the y’s solve for y let f-1(x) be y

2) Find each inverse and graph f(x) and f-1(x). a) f(x) = 3x - 7 b) f(x) = x2 + 2 for x > 2 c) f(x) = log62

3) The population of San Yuan can be modeled by P = 16,500t0.15 , where t is the number of years since 2000. a) Find the inverse model that gives the number of years as a function of the population. Answer : b) What will be the population of San Yuan in 2018? Answer : 25,455 c) In what year will the population reach 50,000? Answer : In the year 3621. I can’t wait!