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Let #f(x) = -1 /(x - 7)# and #g(x) = 8-x^2#, how do you find each of the compositions and domain and range?

Answer 1

The composition of f(g(x)) is -1 / (8 - x^2 - 7) = -1 / (1 - x^2). The composition of g(f(x)) is 8 - (-1/(x - 7))^2 = 8 - 1/(x - 7)^2.

For the domain of f(g(x)), x ≠ ±1. For the range of f(g(x)), the range is all real numbers except 0.

For the domain of g(f(x)), x ≠ 7. For the range of g(f(x)), it is all real numbers.

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Answer 2

Elias Ackerman

The composition of two functions is really just substituting one function as the x value of the other. See example below for detailed explanation.

Example: Find f(g(x))

This means that we must replace x in f(x) for g(x):

=#-1/(8 - x^2 -7)#

= #-1/(1 - x^2)#

= #1/(x + 1 X x - 1)#

The domain on this function would be x cannot be 1 or -1 and the range would be y cannot be 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.