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Answer :
Therefore, the composite function f ∘ g(x) can be expressed as:
f ∘ g(x) = { x^2 + 3 if x^2 - 3 ≤ 6{ 10 if 6 < x^2 - 3 ≤ 10{ x^2 - 9 if x^2 - 3 > 10
If the composite function f ∘ g(x) is given by f ∘ g(x) = If g(x) = x^2 - 3, then what is the function f(x) in terms of x?
The composite function f ∘ g(x) can be computed by first calculating g(x) and then using the result as the input for f(x).
In this case, g(x) is given as x^2 - 3.
To find f(x), we need to apply the function f to the input x. According to the problem, there are three different expressions for f(x), depending on the value of x:
If x ≤ 6, then f(x) = x + 6
If 6 < x ≤ 10, then f(x) = 10
If x > 10, then f(x) = x - 6
So, we need to determine which of these expressions to use based on the value of g(x).
If we substitute g(x) = x^2 - 3 into each of the expressions for f(x), we get:
If g(x) ≤ 6, then f(g(x)) = g(x) + 6 = x^2 + 3
If 6 < g(x) ≤ 10, then f(g(x)) = 10
If g(x) > 10, then f(g(x)) = g(x) - 6 = x^2 - 9
Therefore, the composite function f ∘ g(x) can be expressed as:
f ∘ g(x) = { x^2 + 3 if x^2 - 3 ≤ 6
{ 10 if 6 < x^2 - 3 ≤ 10
{ x^2 - 9 if x^2 - 3 > 10
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