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Domain, range and Domain, range and composite functionscomposite functions

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The domain and range of a function

Remember,

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 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.

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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:

When x = –2, f(x) =

The range of the function is therefore

–8 – 7 = –15

When x = 5, f(x) = 20 – 7 = 13

Since this is a linear function, solve for the smallest and largest values of x:

–15 ≤ f(x) < 13

The function f(x) = 4x – 7 is defined over the domain –2 ≤ x < 5. Find the range of this function.

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Example 1

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Example 2

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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:

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

x

f

x – 3 (x – 3)2

g

g(f)

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Composite functions

g(f(x)) means perform f first and then g.

Compare this with the composite function f(g(x)):

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)).

x

g

x2 x2 – 3

f

f(g)

g(f(x)) is an example of a composite function.

f(f (x)) = (f ◦f )(x)= f(x – 3)= (x – 3) – 3= x – 6

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Composite function machine