Presentation on theme: "The Domain of f is the set of all allowable inputs (x values)"— Presentation transcript:
1 The Domain of f is the set of all allowable inputs (x values) FunctionsA function is an operation performed on an input (x) to produce an output (y = f(x) ).The Domain of f is the set of all allowable inputs (x values)The Range of f is the set of all outputs (y values)DomainRangefxy =f(x)
2 To be well defined a function must · Have a value for each x in the domain· Have only one value for each x in the domaine.g y = f(x) = √(x-1), x is not well defined as if x < 1 we will be trying to square root a negative number.y = f(x) = 1/(x-2), x is not well defined as if x = 2 we will be trying to divide by zero.This is not a function as some x values correspond to two y values.
3 Finding the Range of a function Draw a graph of the function for its given DomainThe Range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph.The Function is f(x) = (x-2)2 +3 , xy = (x-2)2 +3The Range isf(x) ≥ 3Domainy = (x-2)2 +32Range3Link to Inverse FunctionsDomain
4 The Function is f(x) = 3 – 2x , x The Range is f(x) < 3
5 Composite Functions Finding gf(x) g f f(x) g(f(x)) = gf(x) fgNote: gf(x) does not mean g(x) times f(x).f(x)xg(f(x))= gf(x)Note : When finding f(g(x))Replace all the x’s in the rule for the f funcion with the expression for g(x) in a bracket.e.g If f(x) = x2 –2xthen f(x-2) = (x-2)2 – 2(x-2)gf(x) means “g of f of x” i.e g(f(x)) .First we apply the f function.Then the output of the f function becomes the input for the g function.Notice that gf means f first and then g.Example if f(x) = x + 3, x and g(x) = x2 , x thengf(x) = g(f(x)) = g(x + 3) = (x+3)2 , xfg(x) = f(g(x)) = f(x2) = x2 +3, xg2(x) means g(g(x)) = g(x2) = (x2)2 = x4 , xf2(x) means f(f(x)) = f(x+3) = (x+3) + 3 = x + 6 , x
6 Notice that fg and gf are not the same. The Domain of gf is the same as the Domain of f since f is the first function to be applied.The Domain of fg is the same as the Domain of g.For gf to be properly defined the Range (output set) of f must fit inside the Domain (input set) of g.For example if g(x) = √x , x ≥ 0 and f(x) = x – 2, xThen gf would not be well defined as the output of f could be a negative number and this is not allowed as an input for g.However fg is well defined, fg(x) = √x – 2, x ≥ 0.
7 Inverse Functions. Domain of f Range of f The inverse of a function f is denoted by f-1 .The inverse reverses the original function.So if f(a) = b then f-1(b) = a Note: f-1(x) does not mean 1/f(x).bfaf-1Domain of fRange of f= Domain of f-1= Range of f-1
8 One to one FunctionsIf a function is to have an inverse which is also a function then it must be one to one.This means that a horizontal line will never cut the graph more than once.i.e we cannot have f(a) = f(b) if a ≠ b,Two different inputs (x values) are not allowed to give the same output (y value).For instance f(-2) = f(2) = 4y = f(x) = x2 with domain x is not one to one.So the inverse of 4 would have two possibilities : -2 or 2.This means that the inverse is not a function.We say that the inverse function of f does not exist.If the Domain is restricted to x ≥ 0Then the function would be one to one and its inverse would bef-1(x) = √x , x ≥ 0
9 Finding the Rule and Domain of an inverse function Swap over x and yMake y the subjectDomainThe domain of the inverse = the Range of the original.So draw a graph of y = f(x) and use it to find the RangeDrawing the graph of the InverseThe graph of y = f-1(x) is the reflection in y = x of the graph of y = f(x).
10 Find the inverse of the function y = f(x) = (x-2)2 + 3 , x ≥ 2 Example:Find the inverse of the function y = f(x) = (x-2)2 + 3 , x ≥ 2Sketch the graphs of y = f(x) and y = f-1(x) on the same axes showing the relationship between them.DomainThis is the function we considered earlier except that its domain has been restricted to x ≥ 2 in order to make it one-to-one.We know that the Range of f is y ≥ 3 and so the domain of f-1 will be x ≥ 3.RuleSwap x and y to get x = (y-2)2 + 3Now make y the subjectx – 3 = (y-2)2√(x –3) = y-2y = 2 + √(x –3)So Final Answer is:f-1(x) = 2 + √(x –3) , x ≥ 3GraphsReflect in y = x to get the graph of the inverse function.Note: we could also have-√(x –3) = y-2and y = 2 - √(x –3)But this would not fit our function as y must be greater than 2 (see graph)Note: Remember with inverse functions everything swaps over.Input and output (x and y) swap overDomain and Range swap overReflecting in y = x swaps over the coordinates of a point so (a,b) on one graph becomes (b,a) on the other.