Functions A 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) Domain Range f x y =f( x )
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 domain e.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.
Domain y = (x-2) 2 +3 2 The Range is f( x ) ≥ 3 Finding the Range of a function Draw a graph of the function for its given Domain The Range is the set of values on the y-axis for which a horizontal line drawn through that point would cut the graph. Range Domain 3 y = (x-2) 2 +3 The Function is f( x ) = (x-2) 2 +3 , x Link to Inverse Functions
The Function is f( x ) = 3 – 2 x , x The Range is f( x ) < 3
f x g(f( x )) = gf(x) g f( x ) Finding gf(x) Note : gf(x) does not mean g(x) times f(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) = x 2 –2x then f(x-2) = (x-2) 2 – 2(x-2) Composite Functions 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) = x 2 , x then gf(x) = g(f(x)) = g(x + 3) = (x+3) 2 , x fg(x) = f(g(x)) = f(x 2 ) = x 2 +3, x g 2 (x) means g(g(x)) = g(x 2 ) = (x 2 ) 2 = x 4 , x f 2 (x) means f(f(x)) = f(x+3) = (x+3) + 3 = x + 6 , x
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, x Then 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.
Domain of f Range of f = Domain of f -1 = Range of f - 1 Note: f -1 (x) does not mean 1/f(x). Inverse Functions. 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 b f a f -1
One to one Functions If 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) = 4 y = f(x) = x 2 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 ≥ 0 Then the function would be one to one and its inverse would be f -1 (x) = √ x , x ≥ 0
Domain The domain of the inverse = the Range of the original. So draw a graph of y = f(x) and use it to find the Range Finding the Rule and Domain of an inverse function <ul><li>Rule </li></ul><ul><li>Swap over x and y </li></ul><ul><li>Make y the subject </li></ul>Drawing the graph of the Inverse The graph of y = f -1 (x) is the reflection in y = x of the graph of y = f(x).
Example: Find the inverse of the function y = f(x) = (x-2) 2 + 3 , x ≥ 2 Sketch the graphs of y = f(x) and y = f -1 (x) on the same axes showing the relationship between them. Domain This 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. Note: we could also have - √ (x –3) = y-2 and y = 2 - √ (x –3) But this would not fit our function as y must be greater than 2 (see graph) Rule Swap x and y to get x = (y-2) 2 + 3 Now make y the subject x – 3 = (y-2) 2 √ (x –3) = y-2 y = 2 + √ (x –3) So Final Answer is: f -1 (x) = 2 + √ (x –3) , x ≥ 3 Graphs Reflect in y = x to get the graph of the inverse function . Note: Remember with inverse functions everything swaps over. Input and output (x and y) swap over Domain and Range swap over Reflecting in y = x swaps over the coordinates of a point so (a,b) on one graph becomes (b,a) on the other.