Range of a composite function

Hi. I wanted to ask if there was a strategy to find the range of a composite function. For one question I have seen a mark scheme where there is no substitution of the first function into the second function to derive it but I do not understand the logic behind it.

Thank you in advance.

Reply 1

mqb2766

Original post

by TwisterBlade596

You usually take the range of the first tunction and the domain of the second to work out what tthe range of the second is (and hence the composite function). But it helps to see a question/ms youre unsure about.

Reply 2

TwisterBlade596

Here is the question:

Reply 3

mqb2766

Original post

by TwisterBlade596

Here is the question:

They pretty clearly lay it out for you so here you work out the domain/range of g^(-1) then combine the range of g^(-1) with the domain of f and hence determine its range/the range of the composite function.

(edited 10 months ago)

Reply 4

TwisterBlade596

What is the logic behind that? If I know I think it will make it easier for me to apply this technique to other questions too.

Reply 5

mqb2766

Original post

by TwisterBlade596

What is the logic behind that? If I know I think it will make it easier for me to apply this technique to other questions too.

It should be descibed in your textbook, but this composite function is more laboriously written as
f(g^(-1)(x))
so the composite function is evaluated right to left, so first g^(-1) and then f. The output of g^(-1) is the input to f and the resulting output of f represents the range of the composite function. So
domain of g^(-1) -> range of g^(-1) -> combine with domain of f -> range of composite function.

Even more laboriouly you could write the composite function as
G = g^(-1)(x)
y = f(G)
where G is the intermediate variable and its valid values for the composite function are a combination (intersection) of the range of g^(-1) and the domain of f as should be clear?

Reply 6

TwisterBlade596

Original post

by mqb2766

It should be descibed in your textbook, but this composite function is more laboriously written as
f(g^(-1)(x))
so the composite function is evaluated right to left, so first g^(-1) and then f. The output of g^(-1) is the input to f and the resulting output of f represents the range of the composite function. So
domain of g^(-1) -> range of g^(-1) -> combine with domain of f -> range of composite function.
Even more laboriouly you could write the composite function as
G = g^(-1)(x)
y = f(G)
where G is the intermediate variable and its valid values for the composite function are a combination (intersection) of the range of g^(-1) and the domain of f as should be clear?