notes/notes/mat/Funkce/Příklady.md

Date	tags
2022-03-02	mat mat/funkce

Příklady

Určete definiční obory funkcí

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$f_1: y = x^3 - x^2 + 1$ D_{f_1} = \mathbb{R}

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$f_2:y=\frac1{6-4x}$ D_{f_2} = \mathbb{R} - \{1.5\}

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$f_3:y=\frac1{4x^2-9}$ D_{f_3} = \mathbb{R} - \{1.5\}

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$f_4: y = \frac{\sqrt{4-2x}}3$ D_{f_4} = (-\infty; 2\rangle

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$f_5:y=\sqrt{x-2}+\frac1x$ D_{f_5}=\langle2;\infty) - \{0\}

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$f_6:y=\frac{\sqrt{x+3}}{\sqrt{x+7}}$ D_{f_6} = \langle-3;\infty) - \{-7\}

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$f_7:y=\sqrt\frac{x+3}{x+7}$ D_{f_7} = \langle-3;\infty)

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Rozhodněte, zda číslo d náleží H(f_i)

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$d=7$ $f_1:y=3x-4$ $f_1^{-1}: x=\frac{y+4}3$ $D_{f_1^{-1}}=\mathbb{R}$ $H_{f_1}=\mathbb{R}$ $7\in\mathbb{R}$ ✔️

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$d=5$ $f_2:y=-5$ $H_{f_2} = {-5}$ $5 \not\in {-5}$ ❌

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$d=0$ $f_3:y=\frac1{x+3}$ $f_3^{-1} : x=\frac1y-3$ $D_{f_3^{-1}} = \mathbb{R} - {0}$ $0\not\in\mathbb{R}-{0}$ ❌

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$d=-4$ $f_4:y=\frac{x+1}{x-2}$ $f_4^{-1}:2=\frac1y$ $y = \frac{x+1}{x-2}$ $y(x-2)=x+1$ $y(x-2)-x=1$ x-2=\frac{(1+x)}y

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$d=\frac12$ $f_5:y=\frac1{x^2-1}$ $f_5^{-1}: x=\sqrt{\frac1y+1}$ $D_{f_5^{-1}} = H_{F_5} = (-1;\infty)$ $\frac12 \in (-1; \infty)$ ✔️

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Určete číslo b\in\mathbb{R} za předpokladu, že:

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graf funkce f:y=6x+b, prochází bodem $A[0;4]$ $f(0) = 4$ $6*0 + b = 4$ b = 4

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graf funkce f:y=\frac32x+b, prochází bodem $A[-2;9]$ $f(-2) = 9$ $\frac32*(-2)+b=9$ $-3+b=9$ b=12

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Určete číslo a\in\mathbb{R} za předpokladu, že:

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graf funkce f:y=ax+2, prochází bodem $A[-2;6]$ $f(-2) = 6$ $-2a+2=6$ $-2a=4$ $-a=2$ a=-2

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graf funkce f:y=ax-2, prochází bodem $A[1;2]$ $f(1)=2$ $1a-2=2$ a=4

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$$ -5-2x=4x+7 $$ $$ x=-2

$$ 8000+100x=10000+80x $$ $$ x=100

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x + \frac2x - 1 = \frac{x + 6}{x + 2}

x(x+2) + 4 - (x+2) = x + 6 \space | *(x+2)

[x\neq-2]

x^2+2x+4-x-2=x+6

x^2+x+2=x+6

x^2+x-4=x

x^2-4=0

2 = x

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\frac{x+2}{x-1}=\frac{x+6}{x+2} \space | *(x-1)(x+2)

[x\neq1]; [x\neq-2]

(x+2)^2 = (x+6)(x-1)

x^2+4x+4=x^2+5x-6

4x+4=5x-6

x=10

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\frac{2x+1}{x-3}=\frac{4x+2}{2x-1} \space | *(x-3)(2x-1)

[x\neq3]; [x\neq\frac12]

(2x+1)(2x-1)=(4x+2)(x-3)

4x^2-1=4x^2-10x-6 \space|-4x^2

-1=10x-6 \space |+6

5=10x\space|:10

x=0.5

K=\{-\frac12\}

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\frac{x+3}{x-2}=4+\frac5{x-2} \space | *(x-2)

[x\neq2]

x+3=4(x-2)+5

x+3=4x-3 \space |-x

3=3x-3 \space |+3

6=3x \space | :3

x=2

K=\{\emptyset\}

Výsledné x nemá řešení, protože podmínka zakáže výslednou hodnotu (x nesmí být 2, ale x vyšlo 2)

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\frac{x+4}{x+5}=2-\frac1{x+5} \space | *(x+5)

[x\neq-5]

x+4=2(x+5)-1

x+4=2x+9 \space | -x

4=x+9 \space | -9

x=-5

K=\{\emptyset\}

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\frac{x-2}{x+3}=2-\frac5{x+3} \space | * (x+3)

[x\neq-3]

x-2=2(x+3)-5

x-2=2x+1 | -x-1

x=-3

K=\{\emptyset\}

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\frac{x+8}{x+3}=1+\frac5{x+3} \space | *(x+3)

[x\neq-3]

x+8=1(x+3)+5

x+8=x+8 \space | -8

0x=0

K=\{\mathbb{R} - \{-3\}\}

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\frac{x+2}{x+5} = 1-\frac3{x+5} \space | * (x+5)

[x\neq-5]

x+2=x+2 \space | -2-x

0x=0

K=\{\mathbb{R}-\{-5\}\}

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\frac{x-3}{x+2}=1-\frac6{x+2} \space | *(x+2)

[x\neq-2]

x-3=x-4 \space | -x+4

1=0x

K=\{\emptyset\}

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\frac{2x-4}{x-2}=1-\frac2{x-2} \space | *(x-2)

[x\neq2]

2x-4=x-4 | +4

2x=x \space |-x

x=0

K={0}

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\frac{x+3}5 = x

\frac{x+3}5-x=0

x+3-5x=0

-4x+3=0

4x=3

x=\frac34

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7(1-x)-4(x-8)=(2-x)11

7-7x-4x+32=22-11x

-11x+39=22-11x

39\neq22

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\frac{3+2x}2-\frac76=5x-\frac{12x-1}3

9+6x-7=30x-24x+2

6x+2=6x+2

2=2

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x-6=\frac{7x-3}3-\frac{3(x+2)}4

12x-72=28x-12-9x-18

12x-72=27x-30

-72=7x-30

-42=4x

x=-6

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a(a-1)+3=(a+2)(a-2)

aa-a+3=aa-4

-a+3=-4

a=7

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x-(3x-(4x-(3x-1)-1)+3)=-5

x-3x-3+4x-1-3x+1=-5

-x-3=-5

x=2

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x-3(x-5(x-4))=10(x-3)

x-3x+15x-60=10x-30

13x-60=10x-30

3x-60=-30

x=10

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3-\frac32(\frac{5x}6-2)=(1\frac12)x-\frac12(\frac12x-9.5)

3-\frac{15*6x}{2}+\frac62=1.5x-\frac{x}4+\frac{9.5}2

12-30*6x+12=6x-x+19

\frac1{37}=x

🤷

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\frac6{x-5}+1=\frac{2x-4}{x-5}

6+x-5=2x-4

x=5

nemůže být

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\frac{2x+5}{x+2}+\frac1{2+x}=3

2x+5+1=3x+6

2x+6=3x+6

x=0

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\frac{2x+4}{3x-1}=\frac25-\frac{x+5}{1-3x}

\frac{2x+4}{3x-1}=\frac25+\frac{x+5}{3x-1}

2x+4=\frac{2(3x-1)}5 +x+5

x=\frac{6x-2}5+1

x=-3

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\frac{x+7}{x-5}+\frac{x+5}{x-7}=2

(x+7)(x-7)+(x+5)(x-5)=2(x-5)(x-7)

x^2-7^2+x^2-5^2=2(x^2-7x-5x+35)

2x^2-7^2-5^2=2x^2-14x-10x+70

2x^2-49-25=2x^2-24x+70

2x^2-74=2x^2-24x+70

24x=144

x=6

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\frac{17}{x+1}-\frac5{x^2+x}=\frac6x

\frac{17}{x+1}-\frac5{x(x+1)}=\frac6x \space | :x(x+1)

17x-5=6(x+1)

17x-5=6x+6 |+5

11x=11

x=1

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A

1+\log_8x=\log_8(6-x)+2\log_8x

log_8x => $[x>0]$ log_8(6-x) => $[x<6]$ x\in(0;6)

$1=-\log_8x+\log_8(6-x)+2\log_8x$ $1=\log_8\frac{6-x}x$ \log_88=\log_8\frac{6-x}x

1=\log_88

$\log_8(8*x)=\log_8((6-x)x^2)$ $8x=(6-x)x^2$ $x(x^2-6x+8)=0$ x_1=0

x^2-6x+8=0

x(x-2)(x-4)=0

$x_2=2$ x_3=4

x_1=0 => [x>0] => $x_1\ne0$ $x_2=2$ x_3=4

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B

\log z-2=6\log^{-1}z-1

$\log z-2=\frac6{\log z}-1$ \log z-1=\frac6{\log z}

[z>0]

$P=\log z$ [P\ne0]

P-1=\frac6P

$P^2-P=6$ $P^2-P-6=0$ (P-3)(P+2)=0

$P=3$ P=-2

$\log z=3$ z=10^3=1000

$\log z=-2$ z=10^{-2}=0.01=1\%=1/100

$z\in\mathbb{R}-{1}$ z=\{10^3;10^{-2}\}

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C

\log_256=2\log_2(7-x)-\log_2(x+7)+3

\log_256=\log_2((7+x)^2)-\log_2(x+7)+3

3=\log_28

\log_256=\log_2((7+x)^2)-\log_2(x+7)+\log_28

\log_256=\log_2\frac{(7+x)^2*8}{x+7}

56=\frac{(7+x)^2*8}{x+7}

56=\frac{(7^2+14x+x^2)8}{x+7}

56=\frac{(42+14x+x^2)8}{x+7}

56=\frac{8*42+8*14*x+8x^2}{x+7}

56=\frac{336+112x+8x^2}{x+7}

$56x+7*56=336+112x+8x^2$ $56x+392=336+112x+8x^2$ $56x+56=112x+8x^2$ 56=56x+8x^2

\frac25*2=\frac{2*2}5