Suppose that the functions f and g are defined as follows.=fx≠95x, x0=gx−7x9Find the compositions ∘ff and ∘gg.Simplify your answers as much as possible.(Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.)=∘ffx=∘ggx
Question
Suppose that the functions f and g are defined as follows.=fx≠95x, x0=gx−7x9Find the compositions ∘ff and ∘gg.Simplify your answers as much as possible.(Assume that your expressions are defined for all x in the domain of the composition. You do not have to indicate the domain.)=∘ffx=∘ggx
Solution
The compositions of the functions f and g are found by substituting the function into itself.
For ∘ff, we substitute f(x) into f, so we get f(f(x)).
Given f(x) = 5x, we substitute this into f to get f(5x).
Substituting 5x into f(x) = 5x, we get 5(5x) = 25x.
So, ∘ff(x) = 25x.
For ∘gg, we substitute g(x) into g, so we get g(g(x)).
Given g(x) = -7x/9, we substitute this into g to get g(-7x/9).
Substituting -7x/9 into g(x) = -7x/9, we get -7(-7x/9)/9 = 49x/81.
So, ∘gg(x) = 49x/81.
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