Let f(x) = x² and g(x) = sinx for all x  R. Then find the set of all x satisfying (fogogof)(x) = (gogof)(x),
where (fog)(x) = f(g(x)).

To solve the equation \((f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)\) where \(f(x) = x^2\) and \(g(x) = \sin(x)\), let’s break it down step by step.

Understanding the Functions

First, we need to understand what the compositions of these functions mean. The notation \((f \circ g)(x)\) means that we apply \(g\) first and then apply \(f\) to the result of \(g\). Thus:

• \(f(g(x)) = f(\sin(x)) = (\sin(x))^2\)
• \(g(f(x)) = g(x^2) = \sin(x^2)\)

Breaking Down Each Side

Now we calculate each side of the equation:

Left Side: \((f \circ g \circ g \circ f)(x)\)

This means we first apply \(f\), then \(g\), then \(g\) again, and finally \(f\). Let's compute it step-by-step:

• Start with \(x\): \(f(x) = x^2\)
• Apply \(g\): \(g(f(x)) = g(x^2) = \sin(x^2)\)
• Apply \(g\) again: \(g(g(f(x))) = g(\sin(x^2)) = \sin(\sin(x^2))\)
• Finally, apply \(f\):
  • \(f(g(g(f(x)))) = f(\sin(\sin(x^2))) = (\sin(\sin(x^2)))^2\)

Right Side: \((g \circ g \circ f)(x)\)

Now let's compute the right side:

• Start with \(x\): \(f(x) = x^2\)
• Apply \(g\): \(g(f(x)) = g(x^2) = \sin(x^2)\)
• Apply \(g\) again:
  • \(g(g(f(x))) = g(\sin(x^2)) = \sin(\sin(x^2))\)

Setting Up the Equation

Now, we can set the left side equal to the right side:

\((\sin(\sin(x^2)))^2 = \sin(\sin(x^2))\

Solving the Equation

This equation can be simplified. Let \(y = \sin(\sin(x^2))\). The equation now reads:

\(y^2 = y\)

To solve this, we can rearrange it:

\(y^2 - y = 0\)

Factoring gives us:

\(y(y - 1) = 0\)

This leads to two possibilities:

• \(y = 0\)
• \(y = 1\)

Finding Values of x

Now, we’ll solve for \(x\) under each condition.

Case 1: \(y = 0\)

Setting \(\sin(\sin(x^2)) = 0\) implies:

• \(\sin(x^2) = n\pi\) for \(n \in \mathbb{Z}\)

Case 2: \(y = 1\)

Setting \(\sin(\sin(x^2)) = 1\) implies:

• \(\sin(x^2) = \frac{\pi}{2} + 2k\pi\) for \(k \in \mathbb{Z}\)

Conclusion

To summarize, the set of all \(x\) satisfying the original equation is determined by the solutions of the equations derived from \(\sin(\sin(x^2)) = 0\) and \(\sin(\sin(x^2)) = 1\). This leads us to a combination of values of \(x\) based on the periodic nature of the sine function.