If f(x) = sqrt2-x and g(x) = sqrtx, find f circ f and f circ g. | Mathematics I 2075

Mathematics I20755 marksNumerical: function compositionCombination of functions

If $f(x) = \sqrt{2-x}$ and $g(x) = \sqrt{x}$ , find $f \circ f$ and $f \circ g$ . [5]

If $f(x) = \sqrt{2-x}$ and $g(x) = \sqrt{x}$, find $f \circ f$ and $f \circ g$. [5]

$f(x) = \sqrt{2-x}, \qquad g(x) = \sqrt{x}$ Required: $f \circ f$ and $f \circ g$ . --- $(f \circ f)(x) = f(f(x)) = \sqrt{2 - f(x)} = \sqrt{2 - \sqrt{2-x}}$ Domain: Inner root requires: $2 - x \geq 0 \implies x \leq 2$ Outer root requires: $$2 - \sqrt{2-x} \geq 0 \implies \sqrt{2-x} \leq 2 ...

$$f(x) = \sqrt{2-x}, \qquad g(x) = \sqrt{x}$$ Required: $f \circ f$ and $f \circ g$. --- $$(f \circ f)(x) = f(f(x)) = \sqrt{2 - f(x)} = \sqrt{2 - \sqrt{2-x}}$$ Domain: Inner root requires: $$2 - x \geq 0 \implies x \leq 2$$ Outer root requires: $$2 - \sqrt{2-x} \geq 0 \implies \sqrt{2-x} \leq 2 ...

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