فرمولی برای \(f \circ g \circ h\) بنویسید.





$$
f(x) = \frac{x+2}{3-x}\\
g(x) = \frac{x^2}{x^2 +1}\\
h(x) = \sqrt{2-x}
$$

پاسخ

$$
(f \circ g \circ h)(x) = f(g(h(x))) = f(g(\sqrt{2-x})) \\
= f(\frac{ (\sqrt{2-x})^2 }{(\sqrt{2-x})^2 + 1 }) = f(\frac{2-x}{2-x+1})\\
= f(\frac{2-x}{3-x}) = \frac{(\frac{2-x}{3-x})+2}{3-(\frac{2-x}{3-x})}\\
= \frac{ \frac{2-x + 2(3-x)}{3-x} }{ \frac{3(3-x)-(2-x)}{3-x} } = \frac{ \frac{2-x+6-2x}{3-x} }{ \frac{9-3x-2+x}{3-x} } \\
= \frac{ \frac{8-3x}{3-x} }{\frac{7-2x}{3-x}} = \frac{8-3x}{3-x} \cdot \frac{3-x}{7-2x}\\
= \frac{8-3x}{7-2x}
$$