if g:R to R is given by f(x)=xsquare+3,find the function f such that (g circle f)(x)=4xsquare+3

Given the functions \( g: \mathbb{R} \to \mathbb{R} \) defined by \( g(x) = x^2 + 3 \) and the composite function \( (g \circ f)(x) = 4x^2 + 3 \), we need to find the function \( f(x) \).

The composite function \( (g \circ f)(x) \) means that we first apply \( f(x) \) and then apply \( g \) to the result of \( f(x) \). In other words:

\[
(g \circ f)(x) = g(f(x))
\]

We are given:

\[
g(f(x)) = 4x^2 + 3
\]

Since \( g(x) = x^2 + 3 \), substituting \( f(x) \) into \( g(x) \) gives:

\[
g(f(x)) = (f(x))^2 + 3
\]

Thus:

\[
(f(x))^2 + 3 = 4x^2 + 3
\]

Subtracting 3 from both sides:

\[
(f(x))^2 = 4x^2
\]

Taking the square root of both sides:

\[
f(x) = \pm 2x
\]

So, the function \( f(x) \) can be either:

\[
f(x) = 2x \quad \text{or} \quad f(x) = -2x
\]

Both of these functions satisfy the given condition.

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How would the solution change if \( g(x) = x^2 + k \) for some constant \( k \)?
2. What would happen if \( (g \circ f)(x) \) was given as a different quadratic function?
3. Can you find an \( f(x) \) that satisfies \( (f \circ g)(x) = 4x^2 + 3 \) instead?
4. How does the composition of functions relate to their inverses?
5. What if \( f(x) = x^2 + c \) for some constant \( c \)? Would \( g \) have to change?

**Tip:** When dealing with composite functions, always carefully substitute and manipulate the inner function to match the required output.

Explore how to find a function f(x) such that the composite function (g ◦ f)(x) = 4x^2 + 3, given g(x) = x^2 + 3. Learn about composite functions and quadratic equations in this detailed math problem solution.

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