حل أسئلة مراجعة الدرس السادس: العميات على الدوال وتركيب دالتين

أوجد $(f+g)\left(x\right),(f-g)\left(x\right),(f\cdot g)\left(x\right),\left(\frac{f}{g}\right)\left(x\right)$ لكل من الدالتين f(x),g(x) فيما يأتي، ثم اكتب مجال الدالة الناتجة.

44)

$\begin{array}{rl}& f\left(x\right)=x+3\\ & g\left(x\right)=2x^2+4x-6\end{array}$

$\begin{array}{r}(f+g)\left(x\right)=2x^2+5x-3\\ (f-g)\left(x\right)=-2x^2-3x+9\\ (f○g)\left(x\right)=2x^3+10x^2+6x-18\end{array} \mathrm{المجال}: \{x\mid x\in R\}$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{1}{2(x-1)} \mathrm{المجال}: \{x\mid x\ne -3,1,x\in R\}$

45)

$\begin{array}{rl}& f\left(x\right)=4x^2-1\\ & g\left(x\right)=5x-1\end{array}$

$\begin{array}{rl}& (f+g)\left(x\right)=4x^2+5x-2\\ & (f-g)\left(x\right)=4x^2-5x\\ & (f○g)\left(x\right)=20x^3-4x^2-5x+1\end{array} \mathrm{المجال} \{x\mid x\in R\}$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{4x^2-1}{5x-1} \mathrm{المجال}: \{x\mid x\ne \frac{1}{5},x\in R\}$

46)

$\begin{array}{rl}& f\left(x\right)=x^3-2x^2+5\\ & g\left(x\right)=4x^2-3\end{array}$

$$\begin{gathered} \begin{array}{rl}& (f+g)\left(x\right)=x^3+2x^2+2\\ & (f-g)\left(x\right)=x^3-6x^2+8 \mathrm{المجال}: \{x\mid x\in R\}\end{array} \\ (f○g)\left(x\right)=4x^5-8x^4-3x^3+26x^2-15 \end{gathered}$$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{x^3-2x^2+5}{4x^2-3} \mathrm{المجال}: \{x\mid x\ne \pm \frac{\sqrt{3}}{2},x\in R\}$

47)

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

$$\begin{gathered} \begin{array}{rl}& (f+g)\left(x\right)=\frac{x+1}{x^2}\\ & (f-g)\left(x\right)=\frac{x-1}{x^2} \mathrm{المجال}: \{x\mid x\ne 0,x\in R\}\end{array} \\ \begin{array}{rl}& (f○g)\left(x\right)=\frac{1}{x^3}\\ & \left(\frac{f}{g}\right)\left(x\right)=x\end{array} \end{gathered}$$

أوجد $[f\circ g]\left(x\right),[g\circ f]\left(x\right),[f\circ g]\left(2\right)$ لكل دالتين من الدوال الآتية:

48) $f\left(x\right)=4x-11,g\left(x\right)=2x^2-8$

$\begin{array}{rl}& [f\circ g]\left(x\right)=8x^2-43\\ & [g\circ f]\left(x\right)=32x^2-176x+234\\ & [f\circ g]\left(2\right)=8(2{)}^2-43=-11\end{array}$

49) $f\left(x\right)=x^2+2x+8,g\left(x\right)=x-5$

$\begin{array}{rl}& [f\circ g]\left(x\right)=x^2-8x+23\\ & [g\circ f]\left(x\right)=x^2+2x+3\\ & [f\circ g]\left(2\right)=(2{)}^2-16+23=11\end{array}$

50) $f\left(x\right)=x^2-3x+4,g\left(x\right)=x^2$

$\begin{array}{rl}& [f\circ g]\left(x\right)=x^4-3x^2+4\\ & [g\circ f]\left(x\right)=x^4-6x^3+17x^2-24x+16\\ & [f\circ g]\left(2\right)=(2{)}^4-12+4=8\end{array}$

أوجد مجال $f\circ g$ متضمناً أي قيود إذا لزم، ثم أوجد $f\circ g$

51)

$\begin{array}{c}f\left(x\right)=\frac{1}{x-3}\\ g\left(x\right)=2x-6\end{array}$

$[f\circ g]\left(x\right)=\frac{1}{2x-9} \mathrm{المجال}: \left\{x\mid x\ne 3,\frac{{\displaystyle 9}}{{\displaystyle 2}},x\in R\right\}$

52)

$\begin{array}{c}f\left(x\right)=\sqrt{x-2}\\ g\left(x\right)=6x-7\end{array}$

$[f\circ g]\left(x\right)=\sqrt{6x-9} الم\mathrm{جا}ل: [\frac{3}{2},\mathrm{\infty })$