1. Construct the compositions (f ᵒ g) and (g ᵒ f) for two given functions f and g: f (x) = 1/x; g (x) = x^2
1. Construct the compositions (f ᵒ g) and (g ᵒ f) for two given functions f and g: f (x) = 1/x; g (x) = x^2 + 12.
2. Build the composition (f ᵒ g) for two given functions f and g: f (x) = x^3; g (x) = cosine of x.
3. Create the composition (f ᵒ g) for two given functions f and g: f (x) = square root of x; g (x) = 9 + x^3.
4. Generate the composition (f ᵒ g) for two given functions f and g: f (x) = x^2; g (x) = x^2 + 5. f (x) = x^3; g (x) = x^2.
2. Build the composition (f ᵒ g) for two given functions f and g: f (x) = x^3; g (x) = cosine of x.
3. Create the composition (f ᵒ g) for two given functions f and g: f (x) = square root of x; g (x) = 9 + x^3.
4. Generate the composition (f ᵒ g) for two given functions f and g: f (x) = x^2; g (x) = x^2 + 5. f (x) = x^3; g (x) = x^2.
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1. Для начала построим композицию \( (f \circ g) \):\[ (f \circ g)(x) = f(g(x)) \]
\[ (f \circ g)(x) = f(x^2 + 12) \]
\[ (f \circ g)(x) = \frac{1}{x^2 + 12} \]
А теперь построим композицию \( (g \circ f) \):
\[ (g \circ f)(x) = g(f(x)) \]
\[ (g \circ f)(x) = g\left(\frac{1}{x}\right) \]
\[ (g \circ f)(x) = \left(\frac{1}{x}\right)^2 + 12 \]
\[ (g \circ f)(x) = \frac{1}{x^2} + 12 \]
2. Построим композицию \( (f \circ g) \):
\[ (f \circ g)(x) = f(g(x)) \]
\[ (f \circ g)(x) = f(\cos(x)) \]
\[ (f \circ g)(x) = \left(\cos(x)\right)^3 \]
\[ (f \circ g)(x) = \cos^3(x) \]
3. Построим композицию \( (f \circ g) \):
\[ (f \circ g)(x) = f(g(x)) \]
\[ (f \circ g)(x) = f(9 + x^3) \]
\[ (f \circ g)(x) = \sqrt{9 + x^3} \]
4. Построим композицию \( (f \circ g) \):
\[ (f \circ g)(x) = f(g(x)) \]
\[ (f \circ g)(x) = f(x^2 + 5) \]
\[ (f \circ g)(x) = (x^2 + 5)^2 \]
\[ (f \circ g)(x) = x^4 + 10x^2 + 25 \]
Окончательные ответы:
1. Композиция \( (f \circ g)(x) = \frac{1}{x^2 + 12} \), \( (g \circ f)(x) = \frac{1}{x^2} + 12 \).
2. Композиция \( (f \circ g)(x) = \cos^3(x) \).
3. Композиция \( (f \circ g)(x) = \sqrt{9 + x^3} \).
4. Композиция \( (f \circ g)(x) = x^4 + 10x^2 + 25 \).