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Find the domain of
Because we cannot take the square root of a negative number, the domain of g is (−∞,3]. Now we check the domain of the composite function
The domain of this function is (−∞,5]. To find the domain of f∘g, we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since (−∞,3] is a proper subset of the domain of f∘g. This means the domain of f∘g is the same as the domain of g, namely, (−∞,3].
Find the domain of
[−4,0)∪(0,∞)
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function , so we may choose the decomposition that appears to be most expedient.
Write f(x)=√5−x2 as the composition of two functions.
We are looking for two functions, g and h, so f(x)=g(h(x)). To do this, we look for a function inside a function in the formula for f(x). As one possibility, we might notice that the expression 5−x2 is the inside of the square root. We could then decompose the function as
We can check our answer by recomposing the functions.
Write f(x)=43−√4+x2 as the composition of two functions.
Possible answer:
g(x)=√4+x2h(x)=43−xf=h∘g
Access these online resources for additional instruction and practice with composite functions.
| Composite function | (f∘g)(x)=f(g(x)) |
How does one find the domain of the quotient of two functions, fg?
Find the numbers that make the function in the denominator g equal to zero, and check for any other domain restrictions on f and g, such as an even-indexed root or zeros in the denominator.
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