Sagot :
The value of g(x) is (x^3+5) and g(64) = 262149.
According to the statement
we have given that the
f(x)=√x^3 and (fog)(x)=√(x^3+5) and we have to find the value of the g(64).
So, For find the value of g(64), Firstly we have to find the g(x).
So,
We given that
f(x)=√x^3 and (fog)(x)=√(x^3+5)
And here the formula used is
(f o g)(x) = f (g(x))
here (fog)(x)=√(x^3+5) and f(x)=√x^3
From this we get g(x) is (x^3+5)
So,
g(x) = (x^3+5) and
g(64) = ((64)^3+5)
g(64) = 262149.
So, The value of g(x) is (x^3+5) and g(64) = 262149.
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Question:
If f(x)=√x^3 and (fog)(x)=√(x^3+5). Then find the value of g(64).
Learn more about (fog)(x) here
https://brainly.com/question/2328150
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I take this to mean [tex]f(x) = \sqrt{x^3}[/tex] and [tex](f\circ g)(x) = \sqrt x[/tex].
Let's first find the inverse of [tex]f[/tex].
[tex]f\left(f^{-1}(x)\right) = \sqrt{\left(f^{-1}(x)\right)^3} = x \\\\ \implies \left(f^{-1}(x)\right)^3 = x^2 \\\\ \implies f^{-1}(x) = x^{2/3}[/tex]
(Note that [tex]f[/tex] is defined only if [tex]x^3\ge0[/tex], or [tex]x\ge0[/tex].)
Apply the inverse of [tex]f[/tex] to [tex]f\circ g[/tex].
[tex](f\circ g)(x) = f(g(x)) = \sqrt x \\\\ \implies f^{-1}\left(f(g(x))\right) = f^{-1}(\sqrt x) \\\\ \implies g(x) = \left(\sqrt x\right)^{2/3} = \left(x^{1/2}\right)^{2/3} = x^{1/3} = \sqrt[3]{x}[/tex]
Then
[tex]g(64) = \sqrt[3]{64} = \sqrt[3]{4^3} = \boxed{4}[/tex]