Definisi Operasi Fungsi

Misal $f:ℝ\toℝ$ dan $g:ℝ\toℝ$ dua fungsi dan $a\inℝ.$ Maka:

a. $(f\pm g)(x) = f(x) \pm g(x)$

b. $(af)(x) = a f(x)$

c. $(fg)(x) = f(x)g(x)$

d. $(\frac{f}{g})(x)= \frac{f(x)}{g(x)}, g(x) \neq 0$

Catatan:

a. Jika $D(f+g) = D(f) \cap D(g)$

b. Jika $D(af) = D(f)$

c. Jika $D(fg) = D(f) \cap D(g)$ dan,

d. Jika $D(\frac{f}{g})(x)=D(f) \cap D(g), g(x) \neq 0$

Contoh:

Diketahui:
$f(x)=x^2, D(f) = ℝ$ dan $g(x) = 3, D(g) = ℝ.$ Maka:

$$ \begin{aligned} a). (f+g)(x) &= f(x) + g(x)\\ &=x^2 + 3\\ D(f+g)&= ℝ\capℝ = ℝ \end{aligned} $$

$$ \begin{aligned} b). (f-g)(x) &= f(x) - g(x)\\ &=x^2 - 3\\ D(f+g)&= ℝ\capℝ = ℝ \end{aligned} $$

$$ \begin{aligned} c). (5f)(x) &= 5f(x)\\ &=5x^2\\ D(5f)&= ℝ \end{aligned} $$

$$ \begin{aligned} d). (fg)(x) &= f(x) g(x)\\ &=(x^2) (3)\\ &= 3x^2\\ D(fg)&= ℝ\capℝ = ℝ \end{aligned} $$

$$ \begin{aligned} e). (\frac{f}{g})(x) &= \frac{f(x)} {g(x)}\\ &=\frac {x^2} {3}\\ D(\frac{f}{g})&= ℝ\capℝ = ℝ \end{aligned} $$

Silahkan jika ada yang ingin bertanya dan berdiskusi. Terima kasih.