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Answered

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If f(x)=√x^3 and (fog)(x)=√√x, then g(64) =​

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.

Disclaimer: This question was incomplete. Please find the full content below.

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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LammettHash

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]

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