1)zadanie.Wyznacz dziedzine.
a)f(x)=
b)f(x)=
c)f(x)=
d)f(x)=
e)f(x)=
f)f(x)=
g)f(x)=
h)f(x)=
i)f(x)=
j)f(x)=
k)f(x)=
l)f(x)= +
ł)f(x)=
m)f(x)=
2)zadanie. wyznacz dziedzinę i miejsce zerowe
a) f(x)=
b)f(x)=
c)f(x)=
d)f(x)=
e)f(x)=
f)f(X)=
g)f(x)=
f)f(x)=
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1.
a)![f(x)=\frac{1}{x}\\x\neq0\\D_{f}=\mathbb{R}\backslash\{0\} f(x)=\frac{1}{x}\\x\neq0\\D_{f}=\mathbb{R}\backslash\{0\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7Bx%7D%5C%5Cx%5Cneq0%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B0%5C%7D)
b)![f(x)=\frac{1}{x-2}\\x-2\neq0\\x\neq2\\D_{f}=\mathbb{R}\backslash\{2\} f(x)=\frac{1}{x-2}\\x-2\neq0\\x\neq2\\D_{f}=\mathbb{R}\backslash\{2\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7Bx-2%7D%5C%5Cx-2%5Cneq0%5C%5Cx%5Cneq2%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B2%5C%7D)
c)![f(x)=\frac{x}{x+3}\\x+3\neq0\\x\neq-3\\D_{f}=\mathbb{R}\backslash\{-3\} f(x)=\frac{x}{x+3}\\x+3\neq0\\x\neq-3\\D_{f}=\mathbb{R}\backslash\{-3\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%7D%7Bx%2B3%7D%5C%5Cx%2B3%5Cneq0%5C%5Cx%5Cneq-3%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-3%5C%7D)
d)![f(x)=\frac{x+2}{x^{2}-1}\\x^{2}-1\neq0\\x^{2}\neq1\\x\neq-1\wedge x\neq1\\D_{f}=\mathbb{R}\backslash\{-1\ ;\ 1\} f(x)=\frac{x+2}{x^{2}-1}\\x^{2}-1\neq0\\x^{2}\neq1\\x\neq-1\wedge x\neq1\\D_{f}=\mathbb{R}\backslash\{-1\ ;\ 1\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%2B2%7D%7Bx%5E%7B2%7D-1%7D%5C%5Cx%5E%7B2%7D-1%5Cneq0%5C%5Cx%5E%7B2%7D%5Cneq1%5C%5Cx%5Cneq-1%5Cwedge+x%5Cneq1%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-1%5C+%3B%5C+1%5C%7D)
e)![f(x)=\sqrt{x-1}\\x-1\geq0\\x\geq1\\D_{f}=<\ 1\ ;+\infty) f(x)=\sqrt{x-1}\\x-1\geq0\\x\geq1\\D_{f}=<\ 1\ ;+\infty)](https://tex.z-dn.net/?f=f%28x%29%3D%5Csqrt%7Bx-1%7D%5C%5Cx-1%5Cgeq0%5C%5Cx%5Cgeq1%5C%5CD_%7Bf%7D%3D%3C%5C+1%5C+%3B%2B%5Cinfty%29)
f)![f(x)=\sqrt{x+5}\\x+5\geq0\\x\geq-5\\D_{f}=<-5\ ;+\infty) f(x)=\sqrt{x+5}\\x+5\geq0\\x\geq-5\\D_{f}=<-5\ ;+\infty)](https://tex.z-dn.net/?f=f%28x%29%3D%5Csqrt%7Bx%2B5%7D%5C%5Cx%2B5%5Cgeq0%5C%5Cx%5Cgeq-5%5C%5CD_%7Bf%7D%3D%3C-5%5C+%3B%2B%5Cinfty%29)
g)![f(x)=\sqrt{2-x}\\2-x\geq0\\x\leq2\\D_{f}=(-\infty;\ 2> f(x)=\sqrt{2-x}\\2-x\geq0\\x\leq2\\D_{f}=(-\infty;\ 2>](https://tex.z-dn.net/?f=f%28x%29%3D%5Csqrt%7B2-x%7D%5C%5C2-x%5Cgeq0%5C%5Cx%5Cleq2%5C%5CD_%7Bf%7D%3D%28-%5Cinfty%3B%5C+2%3E)
h)![f(x)=\sqrt{x^{2}+4}\\x^{2}+4\geq0\\\forall x\in\mathbb{R}\ x^{2}\geq0\\D_{f}=\mathbb{R} f(x)=\sqrt{x^{2}+4}\\x^{2}+4\geq0\\\forall x\in\mathbb{R}\ x^{2}\geq0\\D_{f}=\mathbb{R}](https://tex.z-dn.net/?f=f%28x%29%3D%5Csqrt%7Bx%5E%7B2%7D%2B4%7D%5C%5Cx%5E%7B2%7D%2B4%5Cgeq0%5C%5C%5Cforall+x%5Cin%5Cmathbb%7BR%7D%5C+x%5E%7B2%7D%5Cgeq0%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D)
i)![f(x)=\frac{1}{\sqrt{x-1}}\\x-1>0\\x>1\\D_{f}=(1\ ;+\infty) f(x)=\frac{1}{\sqrt{x-1}}\\x-1>0\\x>1\\D_{f}=(1\ ;+\infty)](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7Bx-1%7D%7D%5C%5Cx-1%3E0%5C%5Cx%3E1%5C%5CD_%7Bf%7D%3D%281%5C+%3B%2B%5Cinfty%29)
j)![f(x)=\frac{1}{\sqrt{-x}}\\-x>0\\x<0\\D_{f}=(-\infty\ ;\ 0) f(x)=\frac{1}{\sqrt{-x}}\\-x>0\\x<0\\D_{f}=(-\infty\ ;\ 0)](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B-x%7D%7D%5C%5C-x%3E0%5C%5Cx%3C0%5C%5CD_%7Bf%7D%3D%28-%5Cinfty%5C+%3B%5C+0%29)
k)![f(x)=\frac{1}{\sqrt{x^{2}+1}}\\x^{2}+1>0\\\forall\ x\in\mathbb{R}\ x^{2}+1>0\\D_{f}=\mathbb{R} f(x)=\frac{1}{\sqrt{x^{2}+1}}\\x^{2}+1>0\\\forall\ x\in\mathbb{R}\ x^{2}+1>0\\D_{f}=\mathbb{R}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%5E%7B2%7D%2B1%7D%7D%5C%5Cx%5E%7B2%7D%2B1%3E0%5C%5C%5Cforall%5C+x%5Cin%5Cmathbb%7BR%7D%5C+x%5E%7B2%7D%2B1%3E0%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D)
l)![f(x)=\frac{1}{x+1}+\frac{1}{x+2}\\x+1\neq0\wedge x+2\neq0\\x\neq-1\wedge x\neq-2\\D_{f}=\mathbb{R}\backslash\{-2\ ;-1\} f(x)=\frac{1}{x+1}+\frac{1}{x+2}\\x+1\neq0\wedge x+2\neq0\\x\neq-1\wedge x\neq-2\\D_{f}=\mathbb{R}\backslash\{-2\ ;-1\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7Bx%2B1%7D%2B%5Cfrac%7B1%7D%7Bx%2B2%7D%5C%5Cx%2B1%5Cneq0%5Cwedge+x%2B2%5Cneq0%5C%5Cx%5Cneq-1%5Cwedge+x%5Cneq-2%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-2%5C+%3B-1%5C%7D)
ł)![f(x)=\frac{1}{x(x+4)}\\x(x+4)\neq0\\x\neq0\wedge x\neq-4\\D_{f}=\mathbb{R}\backslash\{-4\ ;\ 0\} f(x)=\frac{1}{x(x+4)}\\x(x+4)\neq0\\x\neq0\wedge x\neq-4\\D_{f}=\mathbb{R}\backslash\{-4\ ;\ 0\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B1%7D%7Bx%28x%2B4%29%7D%5C%5Cx%28x%2B4%29%5Cneq0%5C%5Cx%5Cneq0%5Cwedge+x%5Cneq-4%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-4%5C+%3B%5C+0%5C%7D)
m)![f(x)=\frac{x+2}{(x+5)(x+6)}\\(x+5)(x+6)\neq0\\x\neq-5\wedge x\neq-6\\D_{f}=\mathbb{R}\backslash\{-6\ ;-5\} f(x)=\frac{x+2}{(x+5)(x+6)}\\(x+5)(x+6)\neq0\\x\neq-5\wedge x\neq-6\\D_{f}=\mathbb{R}\backslash\{-6\ ;-5\}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%2B2%7D%7B%28x%2B5%29%28x%2B6%29%7D%5C%5C%28x%2B5%29%28x%2B6%29%5Cneq0%5C%5Cx%5Cneq-5%5Cwedge+x%5Cneq-6%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-6%5C+%3B-5%5C%7D)
2.
a)![f(x)=\frac{x^{2}-9}{\sqrt{x}}\\x>0\Rightarrow D_{f}=(0;+\infty)\\x^{2}-9=0\\(x-3)(x+3)=0\\x=3 f(x)=\frac{x^{2}-9}{\sqrt{x}}\\x>0\Rightarrow D_{f}=(0;+\infty)\\x^{2}-9=0\\(x-3)(x+3)=0\\x=3](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E%7B2%7D-9%7D%7B%5Csqrt%7Bx%7D%7D%5C%5Cx%3E0%5CRightarrow+D_%7Bf%7D%3D%280%3B%2B%5Cinfty%29%5C%5Cx%5E%7B2%7D-9%3D0%5C%5C%28x-3%29%28x%2B3%29%3D0%5C%5Cx%3D3)
Odrzucamy x=-3, bo dziedzina uwzględnia tylko liczby dodatnie.
b)![f(x)=\frac{x^{2}-9}{x(x-3)}\\D_{f}=\mathbb{R}\backslash\{0\ ;\ 3\}\\x^{2}-9=0\\x=-3 f(x)=\frac{x^{2}-9}{x(x-3)}\\D_{f}=\mathbb{R}\backslash\{0\ ;\ 3\}\\x^{2}-9=0\\x=-3](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E%7B2%7D-9%7D%7Bx%28x-3%29%7D%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B0%5C+%3B%5C+3%5C%7D%5C%5Cx%5E%7B2%7D-9%3D0%5C%5Cx%3D-3)
c)![f(x)=\frac{x^{2}-4}{\sqrt{x-5}}\\D_{f}=(5\ ;+\infty)\\x^{2}-4=0\\x_{1}=-2\vee x_{2}=-2\\x_{1},x_{2}\ \notin D_{f}\\brak\ miejsc\ zerowych f(x)=\frac{x^{2}-4}{\sqrt{x-5}}\\D_{f}=(5\ ;+\infty)\\x^{2}-4=0\\x_{1}=-2\vee x_{2}=-2\\x_{1},x_{2}\ \notin D_{f}\\brak\ miejsc\ zerowych](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E%7B2%7D-4%7D%7B%5Csqrt%7Bx-5%7D%7D%5C%5CD_%7Bf%7D%3D%285%5C+%3B%2B%5Cinfty%29%5C%5Cx%5E%7B2%7D-4%3D0%5C%5Cx_%7B1%7D%3D-2%5Cvee+x_%7B2%7D%3D-2%5C%5Cx_%7B1%7D%2Cx_%7B2%7D%5C+%5Cnotin+D_%7Bf%7D%5C%5Cbrak%5C+miejsc%5C+zerowych)
d)![f(x)=\frac{x^{2}-16}{\sqrt{x-4}}\\D_{f}=(4;+\infty)\\x^{2}-16=0\\brak\ miejsc\ zerowych f(x)=\frac{x^{2}-16}{\sqrt{x-4}}\\D_{f}=(4;+\infty)\\x^{2}-16=0\\brak\ miejsc\ zerowych](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E%7B2%7D-16%7D%7B%5Csqrt%7Bx-4%7D%7D%5C%5CD_%7Bf%7D%3D%284%3B%2B%5Cinfty%29%5C%5Cx%5E%7B2%7D-16%3D0%5C%5Cbrak%5C+miejsc%5C+zerowych)
Jak wyżej - miejsca zerowe nie należą do dziedziny.
e)![f(x)=\frac{x-2}{x^{2}-5}\\D_{f}=\mathbb{R}\backslash\{-\sqrt{5}\ ;\ \sqrt{5}\}\\x-2=0\\x=2 f(x)=\frac{x-2}{x^{2}-5}\\D_{f}=\mathbb{R}\backslash\{-\sqrt{5}\ ;\ \sqrt{5}\}\\x-2=0\\x=2](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx-2%7D%7Bx%5E%7B2%7D-5%7D%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-%5Csqrt%7B5%7D%5C+%3B%5C+%5Csqrt%7B5%7D%5C%7D%5C%5Cx-2%3D0%5C%5Cx%3D2)
f)![f(x)=\frac{x^{2}-2}{(x-3)(x+2)}\\D{f}=\mathbb{R}\backslash\{-2\ ;\ 3\}\\x^{2}-2=0\\x_{1}=-\sqrt{2}\vee x_{2}=\sqrt{2} f(x)=\frac{x^{2}-2}{(x-3)(x+2)}\\D{f}=\mathbb{R}\backslash\{-2\ ;\ 3\}\\x^{2}-2=0\\x_{1}=-\sqrt{2}\vee x_{2}=\sqrt{2}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E%7B2%7D-2%7D%7B%28x-3%29%28x%2B2%29%7D%5C%5CD%7Bf%7D%3D%5Cmathbb%7BR%7D%5Cbackslash%5C%7B-2%5C+%3B%5C+3%5C%7D%5C%5Cx%5E%7B2%7D-2%3D0%5C%5Cx_%7B1%7D%3D-%5Csqrt%7B2%7D%5Cvee+x_%7B2%7D%3D%5Csqrt%7B2%7D)
g)![f(x)=\frac{x^{2}-1}{\sqrt{x(x+3)}}\\x(x+3)>0\\D_{f}=(-\infty\ ;-3)\cup(0\ ;+\infty)\\x^{2}-1=0\\x=1 f(x)=\frac{x^{2}-1}{\sqrt{x(x+3)}}\\x(x+3)>0\\D_{f}=(-\infty\ ;-3)\cup(0\ ;+\infty)\\x^{2}-1=0\\x=1](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%5E%7B2%7D-1%7D%7B%5Csqrt%7Bx%28x%2B3%29%7D%7D%5C%5Cx%28x%2B3%29%3E0%5C%5CD_%7Bf%7D%3D%28-%5Cinfty%5C+%3B-3%29%5Ccup%280%5C+%3B%2B%5Cinfty%29%5C%5Cx%5E%7B2%7D-1%3D0%5C%5Cx%3D1)
x=-1 odrzucamy, ponieważ nie należy do dziedziny.
f)![f(x)=\frac{x+3}{|x|+3}\\\forall x\in\mathbb{R}\ |x|\geq0\\D_{f}=\mathbb{R}\\x+3=0\\x=-3 f(x)=\frac{x+3}{|x|+3}\\\forall x\in\mathbb{R}\ |x|\geq0\\D_{f}=\mathbb{R}\\x+3=0\\x=-3](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7Bx%2B3%7D%7B%7Cx%7C%2B3%7D%5C%5C%5Cforall+x%5Cin%5Cmathbb%7BR%7D%5C+%7Cx%7C%5Cgeq0%5C%5CD_%7Bf%7D%3D%5Cmathbb%7BR%7D%5C%5Cx%2B3%3D0%5C%5Cx%3D-3)