Domain range and composite functions 1 of 8

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Domain, range and composite functions 1 of 8 © Boardworks Ltd 2012

Domain, range and composite functions 1 of 8 © Boardworks Ltd 2012

The domain and range of a function Remember, The domain of a function is

The domain and range of a function Remember, The domain of a function is the set of values to which the function can be applied. The range of a function is the set of all possible output values. A function is only fully defined if we are given both: the rule that defines the function, for example f(x) = x – 4. the domain of the function, for example the set {1, 2, 3, 4}. Given the rule f(x) = x – 4 and the domain {1, 2, 3, 4} we can find the range: {– 3, – 2, – 1, 0} 2 of 8 © Boardworks Ltd 2012

The domain and range of a function It is more common for a function

The domain and range of a function It is more common for a function to be defined over a continuous interval, rather than a set of discrete values. For example: The function f(x) = 4 x – 7 is defined over the domain – 2 ≤ x < 5. Find the range of this function. Since this is a linear function, solve for the smallest and largest values of x: When x = – 2, f(x) = – 8 – 7 = – 15 When x = 5, f(x) = 20 – 7 = 13 The range of the function is therefore – 15 ≤ f(x) < 13 3 of 8 © Boardworks Ltd 2012

Example 1 4 of 8 © Boardworks Ltd 2012

Example 1 4 of 8 © Boardworks Ltd 2012

Example 2 5 of 8 © Boardworks Ltd 2012

Example 2 5 of 8 © Boardworks Ltd 2012

Composite functions Suppose we have two functions defined for all real numbers: f(x) =

Composite functions Suppose we have two functions defined for all real numbers: f(x) = x – 3 g(x) = x 2 We can combine these two functions by applying f and then applying g as follows: f g x x– 3 (x – 3)2 g(f) Since we are applying g to f(x), this can be written as g(f(x)) or more simply as (g◦f)(x). So: g(f(x)) = (x – 3)2 6 of 8 © Boardworks Ltd 2012

Composite functions g(f(x)) is an example of a composite function. g(f(x)) means perform f

Composite functions g(f(x)) is an example of a composite function. g(f(x)) means perform f first and then g. Compare this with the composite function f(g(x)): g f x x 2 – 3 f(g) It is also possible to form a composite function by applying the same function twice. For example, if we apply the function f to f(x), we have f(f(x)). f(f (x)) = (f ◦f )(x) = f(x – 3) = (x – 3) – 3 =x– 6 7 of 8 © Boardworks Ltd 2012

Composite function machine 8 of 8 © Boardworks Ltd 2012

Composite function machine 8 of 8 © Boardworks Ltd 2012