Let #f(x)= 1-x^3# and #g(x)= 1/x#, how do you find each of the compositions?

Answer 1

To find the composition of (f) and (g), denoted as ((f \circ g)(x)), and the composition of (g) and (f), denoted as ((g \circ f)(x)), follow these steps:

  1. Finding (f \circ g): ((f \circ g)(x) = f(g(x))) Substitute (g(x)) into (f(x)) to get: ((f \circ g)(x) = f(1/x)) Then, replace (x) in (f(x)) with (1/x): ((f \circ g)(x) = 1 - (1/x)^3) Simplify the expression: ((f \circ g)(x) = 1 - \frac{1}{x^3})

  2. Finding (g \circ f): ((g \circ f)(x) = g(f(x))) Substitute (f(x)) into (g(x)) to get: ((g \circ f)(x) = g(1-x^3)) Then, replace (x) in (g(x)) with (1-x^3): ((g \circ f)(x) = \frac{1}{1-x^3})

Therefore, the compositions are: ((f \circ g)(x) = 1 - \frac{1}{x^3}) ((g \circ f)(x) = \frac{1}{1-x^3})

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Answer 2

Think of the #x# as just a place holder for something to get:

#(f*g)(x) = f(g(x)) = 1-(g(x))^3 = 1-(1/x)^3 = 1-1/x^3#

#(g*f)(x) = g(f(x)) = 1/f(x) = 1/(1-x^3)#

An equation like #f(x) = 1 - x^3# basically describes what you do to any thing to turn it into #f# of that thing.
You can replace #x# with any number or expression and the equation remains true.
So to find #(f*g)# just substitute the expression for #g(x)# into the equation for #f(x)# to get:
#(f*g)(x) = f(g(x)) = 1-(g(x))^3 = 1-(1/x)^3 = 1-1/x^3#

Similarly,

#(g*f)(x) = g(f(x)) = 1/f(x) = 1/(1-x^3)#
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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.

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.

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.

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.

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