أوجد $\left(\frac{f}{g}\right)\left(x\right)،(f\cdot g)\left(x\right)،(f-g)\left(x\right)،(f+g)\left(x\right)$ ، لكل زوج من الدوال الآتية، ثم حدّد مجال الدالة الناتجة:

$\begin{array}{r}f\left(x\right)=\frac{x}{x+1}\\ g\left(x\right)=x^2-1\end{array}$

$\begin{array}{rl}(f+g)\left(x\right)& =f\left(x\right)+g\left(x\right)\\ & =\frac{x}{x+1}+x^2-1\end{array}$

المجال: $\{x\mid x\ne -1,x\in \mathrm{R}\}$

$\begin{array}{rl}(f-g)\left(x\right)& =f\left(x\right)-g\left(x\right)\\ & =\frac{x}{x+1}-x^2+1\end{array}$

المجال: $\{x\mid x\ne -1,x\in \mathrm{R}\}$

$\begin{array}{rl}(f\cdot g)\left(x\right)& =f\left(x\right)\cdot g\left(x\right)\\ & =\frac{x}{x+1}\cdot (x-1)(x+1)\\ & =x(x-1)\end{array}$

المجال: $\{x\mid x\ne -1,x\in \mathrm{R}\}$

$\begin{array}{rl}\left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(x\right)=\frac{f\left(x\right)}{\mathrm{g}\left(x\right)}=& \frac{x}{x+1}÷(x^2-1)\\ & =\frac{x}{x+1}\cdot \frac{1}{x^2-1}\end{array}$

المجال: $\{x\mid x\ne -1,x\ne 1,x\in \mathrm{R}\}$

حل أسئلة مراجعة تراكمية

استعمل التمثيل البياني للدالة (f (x أدناه لإيجاد كلّ مما يأتي:

54) $f(-2)\cdot \underset{x\to -2}{lim}f\left(x\right)$

1-،1-

55) $f\left(0\right)،\underset{x\to 0}{lim}f\left(x\right)$

0، غير معرفة.

56) $f\left(3\right)،\underset{x\to 3}{lim}f\left(x\right)$

4,2

أوجد $\left(\frac{f}{g}\right)\left(x\right)،(f\cdot g)\left(x\right)،(f-g)\left(x\right)،(f+g)\left(x\right)$، لكل زوج من الدوال الآتية، ثم حدّد مجال الدالة الناتجة:

57)

$\begin{array}{r}f\left(x\right)=x^2-2x\\ g\left(x\right)=x+9\end{array}$

$\begin{array}{rl}(f+g)\left(x\right)& =f\left(x\right)+g\left(x\right)\\ & =x^2-2x+x+9\\ & =x^2-x+9\end{array}$

المجال: $R\text{أو}(-\mathrm{\infty },\mathrm{\infty })$

$\begin{array}{rl}(f-g)\left(x\right)& =f\left(x\right)-g\left(x\right)\\ & =x^2-2x-x-9\\ & =x^2-3x-9\end{array}$

المجال: $R\text{أو}(-\mathrm{\infty },\mathrm{\infty })$

$\begin{array}{rl}(f\cdot g)\left(x\right)& =f\left(x\right)\cdot g\left(x\right)\\ & =(x^2-2x)\cdot (x+9)\\ & =x^3+9x^2-2x^2-18x\\ & =x^3+7x^2-18x\end{array}$

المجال: $R\text{أو}(-\mathrm{\infty },\mathrm{\infty })$

$\left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(x\right)=\frac{f\left(x\right)}{\mathrm{g}\left(x\right)}=\frac{x^2-2x}{x+9}$

المجال: $\{x\mid x\ne -9,x\in \mathrm{R}\}$

58)

$\begin{array}{r}f\left(x\right)=\frac{x}{x+1}\\ g\left(x\right)=x^2-1\end{array}$

$\begin{array}{rl}(f+g)\left(x\right)& =f\left(x\right)+g\left(x\right)\\ & =\frac{x}{x+1}+x^2-1\end{array}$

المجال: $\{x\mid x\ne -1,x\in \mathrm{R}\}$

$\begin{array}{rl}(f-g)\left(x\right)& =f\left(x\right)-g\left(x\right)\\ & =\frac{x}{x+1}-x^2+1\end{array}$

المجال: $\{x\mid x\ne -1,x\in \mathrm{R}\}$

$\begin{array}{rl}(f\cdot g)\left(x\right)& =f\left(x\right)\cdot g\left(x\right)\\ & =\frac{x}{x+1}\cdot (x-1)(x+1)\\ & =x(x-1)\end{array}$

المجال: $\{x\mid x\ne -1,x\in \mathrm{R}\}$

$\begin{array}{rl}\left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(x\right)=\frac{f\left(x\right)}{\mathrm{g}\left(x\right)}=& \frac{x}{x+1}÷(x^2-1)\\ & =\frac{x}{x+1}\cdot \frac{1}{x^2-1}\end{array}$

المجال: $\{x\mid x\ne -1,x\ne 1,x\in \mathrm{R}\}$