أوجد مجال كل دالة من الدوال الآتية:

$f\left(x\right)=x^2-\sqrt{2}$

$\{x\mid x\in R\}$

$f\left(x\right)=\frac{1}{x^2-16}$

$\{x\mid x\ne \pm 4,x\in R\}$

$f\left(x\right)=\sqrt{3x+18}$

$\{x\mid x\ge -6,x\in R\}$

حل أسئلة مراجعة تراكمية

أوجد القيم المطلوبة لكل دالة مما يأتي:

69) $g\left(x\right)=x^2-10x+3$

a) g(2)

$\begin{array}{rl}g\left(2\right)& =2^2-10\left(2\right)+3\\ & =4-20+3=-13\end{array}$

b) g(-4x)

$\begin{array}{rl}g(-4x)=& (-4x{)}^2-10(-4x)+3\\ =& 16x^2+40x+3\end{array}$

c) g(1+3n)

$\begin{array}{rl}g(1+3n)& =(1+3n{)}^2-10(1+3n)+3\\ & =1+9n^2+6n-10-30n+3\\ & =9n^2-24n-6\end{array}$

70) $p\left(x\right)=\frac{2x^3+2}{x^2-2}$

a) p(3)

$\begin{array}{rl}p\left(3\right)& =\frac{2(3{)}^3+2}{(3{)}^2-2}\\ & =\frac{56}{7}=8\end{array}$

b) p(x²)

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

c) p(x+1)

$\begin{array}{rl}p(x& +1)=\frac{2(x+1{)}^3+2}{(x+1{)}^2-2}\\ & =\frac{2(x^3+3x^2+3x+1)+2}{x^2+2x+1-2}\\ & =\frac{2x^3+6x^2+6x+4}{x^2+2x-1}\end{array}$

71) $h\left(x\right)=2x^2+4x-7$

a) h(-9)

$\begin{array}{rl}h(-9)& =2(-9{)}^2+4(-9)-7\\ & =2\times 81-36-7\\ & =162-43=119\end{array}$

b) h(3x)

$\begin{array}{rl}h\left(3x\right)=& 2(3x{)}^2+4(3x)-7\\ =& 18x^2+12x-7\end{array}$

c) h(2+m)

$\begin{array}{rl}h(2+m)& =2(2+m{)}^2+4(2+m)-7\\ & =2(4+4m+m^2)+8+4m-7\\ & =8+8m+2m^2+8+4m-7\\ & =2m^2+12m+9\end{array}$

أوجد مجال كل دالة من الدوال الآتية:

72) $f\left(x\right)=x^2-\sqrt{2}$

$\{x\mid x\in R\}$

73) $f\left(x\right)=\frac{1}{x^2-16}$

$\{x\mid x\ne \pm 4,x\in R\}$

74) $f\left(x\right)=\sqrt{3x+18}$

$\{x\mid x\ge -6,x\in R\}$

بسّط كلاً مما يأتي:

75) ${27}^{\frac{1}{3}}$

3

76) ${64}^{\frac{5}{6}}$

32

77) ${49}^{-\frac{1}{2}}$

$\frac{1}{7}$

78) ${16}^{-\frac{3}{4}}$

$\frac{1}{8}$

79) ${25}^{\frac{3}{2}}$

125

80) ${36}^{-\frac{3}{2}}$

$\frac{1}{216}$