Domain, range and composite functions

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}

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) = 4x – 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 With many-to-one functions such as f(x) = sin(x) or f(x) = x2, solving for the smallest and largest values of x may not give the range of the function. In these cases it is more important to sketch the function. When x = 5, f(x) = 20 – 7 = 13 The range of the function is therefore –15 ≤ f(x) < 13

Example 1 Change the domain and function and observe the change in the range.

Example 2 This example shows how the range of a quadratic function depends not only on the given domain but also on the maximum and minimum value of the function. This can be found by writing the function in vertex form.

Composite functions Suppose we have two functions defined for all real numbers: f(x) = x – 3 g(x) = x2 We can combine these two functions by applying f and then applying g as follows: x f x – 3 (x – 3)2 g 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

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)): x g x2 x2 – 3 f 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

Composite function machine