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functions - Coggle Diagram
functions
injective function
injective functions - a function is said to be injective if every output in b has a unique input in a
it is not necessary that every element in the codomain has a corresponding element in the domain
horizontal line test - if it crosses the line at most once then it is said to be injective
surjective functions
surjective - every element of the codomain has at least one element of the domain
no element in the codomain is unused ,
hence the range = domain
horizontal line test - if the function is surjective , then every horizontal line drawn must intersect the graph at least once
function
each x value has a unique y value
in the mapping of a function, only one arrow comes from each element of the domain
each element of the domain is operated by the function
bijective functions
bijective function is both surjective and injective
a function is bijective if for every element in the b has an exactly 1 element in a such that f(x) = y
sketching different graphs
quadratic - general form of a quadratic function is ax2 + bx +c where a,b,c are constants and a is not = to 0
when a is greater than 0 the graph would be a u shape
lowest point on the curve = minimum point
when is is less then 0 the graph would be a n type shape
highest point on curve is maximum point
to draw a rough sketch you need sketch you need
point of intersection on x and y axis
turning points
these are found be using the completed square form
for the function y = k(x - p)"2 + q
the turning point is (p,q)
cubic functions - general function is ax3 + bx2 +cx +d which generally consists of a smooth curve with two turning points
when a is greater than 0 the graph is rising from left to right
when a is less than 0 the graph is falling from left to right
example - y = x(x+1)(x-2)
1) y = x3 - x2 - 2x , multiply brackets out.
2) let y = 0 , (x+0)(x+1)(x-2) = 0 , x = 0 . x=-1, x=2. the curve crosses the x axis at (0,0) , (-1,0) , (2,0)
3) let x = 0 to solve y , y= 0 - 0 - 2(0) , the line crosses the y axis at the point (0,0)
linear - the graph of a linear function can be found by find the points that intersects the x and y axis
example - y = 2x -4
let x =0 , y = 2(0) - 4 , y =-4
let y = 0 , 2x - 4 = 0 , x = 2
points are (0,-4) and (2,0)
vertical line tests
if a vertical line is drawn anywhere along the graph , and intersects the graph at more than one point then it is not a function
composition of functions
in general is f and g are two functions then fg(x) is not equal to gf(x)
inverse functions - how to find an inverse function
1) f(x) = 3x - 2 - function , f(x) = y
2) y = 3x - 2
3) 3x = y +2 - isolate x
4) x = y+2 all over 3 - divide by 3 to get x on its own
5)f'(x) = x +2 all over 3 - replace x with y to get inverse function
