Find the value of:(𝑓𝑔)(2)−(𝑔𝑓(4))(fg)(2)−( fg​ (4))left parenthesis, f, g, right parenthesis, left parenthesis, 2, right parenthesis, minus, left parenthesis, start fraction, g, divided by, f, end fraction, left parenthesis, 4, right parenthesis, right parenthesisGiven the functions:𝑓(𝑥)=1𝑥𝑔(𝑥)=3𝑥+1f(x)g(x)​ = x1​ =3x+1​

Para resolver la expresión $(fg)(2) - (gf(4))$ dados los funciones $f(x) = \frac{1}{x}$ y $g(x) = 3x + 1$, sigue estos pasos:

1. Calcular $f(2)$: $f(2) = \frac{1}{2}$

2. Calcular $g(2)$: $g(2) = 3(2) + 1 = 6 + 1 = 7$

3. Calcular $(fg)(2)$: $(fg)(2) = f(2) \cdot g(2) = \frac{1}{2} \cdot 7 = \frac{7}{2}$

4. Calcular $f(4)$: $f(4) = \frac{1}{4}$

5. Calcular $g(f(4))$: $g(f(4)) = g\left(\frac{1}{4}\right) = 3\left(\frac{1}{4}\right) + 1 = \frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4}$

6. Calcular $(gf(4))$: $(gf(4)) = g(f(4)) = \frac{7}{4}$

7. Calcular la expresión completa $(fg)(2) - (gf(4))$: $(fg)(2) - (gf(4)) = \frac{7}{2} - \frac{7}{4}$

8. Simplificar la resta: $\frac{7}{2} - \frac{7}{4} = \frac{14}{4} - \frac{7}{4} = \frac{14 - 7}{4} = \frac{7}{4}$

Por lo tanto, el valor de $(fg)(2) - (gf(4))$ es $\frac{7}{4}$.