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10.2E: Mazoezi

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    Mazoezi hufanya kamili

    Zoezi\(\PageIndex{19}\) Find and Evaluate Composite Functions

    Katika mazoezi yafuatayo, tafuta

    1. \((f \circ g)(x)\)
    2. \((g \circ f)(x)\)
    3. \((f \cdot g)(x)\)
    1. \(f(x)=4 x+3\)na\(g(x)=2 x+5\)
    2. \(f(x)=3 x-1\)na\(g(x)=5 x-3\)
    3. \(f(x)=6 x-5\)na\(g(x)=4 x+1\)
    4. \(f(x)=2 x+7\)na\(g(x)=3 x-4\)
    5. \(f(x)=3 x\)na\(g(x)=2 x^{2}-3 x\)
    6. \(f(x)=2 x\)na\(g(x)=3 x^{2}-1\)
    7. \(f(x)=2 x-1\)na\(g(x)=x^{2}+2\)
    8. \(f(x)=4 x+3\)na\(g(x)=x^{2}-4\)
    Jibu

    1.

    1. \(8x+23\)
    2. \(8x+11\)
    3. \(8 x^{2}+26 x+15\)

    3.

    1. \(24x+1\)
    2. \(24x-19\)
    3. \(24x^{2}+19x-5\)

    5.

    1. \(6 x^{2}-9 x\)
    2. \(18 x^{2}-9 x\)
    3. \(6 x^{3}-9 x^{2}\)

    7.

    1. \(2 x^{2}+3\)
    2. \(4 x^{2}-4 x+3\)
    3. \(2 x^{3}-x^{2}+4 x-2\)
    Zoezi\(\PageIndex{20}\) Find and Evaluate Composite Functions

    Katika mazoezi yafuatayo, pata maadili yaliyoelezwa.

    1. Kwa kazi\(f(x)=2 x^{2}+3\) na\(g(x)=5x-1\), tafuta
      1. \((f \circ g)(-2)\)
      2. \((g \circ f)(-3)\)
      3. \((f \circ f)(-1)\)
    2. Kwa kazi\(f(x)=5 x^{2}-1\) na\(g(x)=4x−1\), tafuta
      1. \((f \circ g)(1)\)
      2. \((g \circ f)(-1)\)
      3. \((f \circ f)(2)\)
    3. Kwa kazi\(f(x)=2x^{3}\) na\(g(x)=3x^{2}+2\), tafuta
      1. \((f \circ g)(-1)\)
      2. \((g \circ f)(1)\)
      3. \((g \circ g)(1)\)
    4. Kwa kazi\(f(x)=3 x^{3}+1\) na\(g(x)=2 x^{2}=3\), tafuta
      1. \((f \circ g)(-2)\)
      2. \((g \circ f)(-1)\)
      3. \((g \circ g)(1)\)
    Jibu

    1.

    1. \(245\)
    2. \(104\)
    3. \(53\)

    3.

    1. \(250\)
    2. \(14\)
    3. \(77\)
    Zoezi\(\PageIndex{21}\) Determine Whether a Function is One-to-One

    Katika mazoezi yafuatayo, onyesha kama seti ya jozi zilizoamriwa inawakilisha kazi na ikiwa ni hivyo, ni kazi moja kwa moja.

    1. \(\begin{array}{l}{\{(-3,9),(-2,4),(-1,1),(0,0)}, {(1,1),(2,4),(3,9) \}}\end{array}\)
    2. \(\begin{array}{l}{\{(9,-3),(4,-2),(1,-1),(0,0)}, {(1,1),(4,2),(9,3) \}}\end{array}\)
    3. \(\begin{array}{l}{\{(-3,-5),(-2,-3),(-1,-1)}, {(0,1),(1,3),(2,5),(3,7) \}}\end{array}\)
    4. \(\begin{array}{l}{\{(5,3),(4,2),(3,1),(2,0)}, {(1,-1),(0,-2),(-1,-3) \}}\end{array}\)
    Jibu

    1. Kazi; si moja kwa moja

    3. Kazi moja kwa moja

    Zoezi\(\PageIndex{22}\) Determine Whether a Function is One-to-One

    Katika mazoezi yafuatayo, onyesha kama kila grafu ni grafu ya kazi na ikiwa ni hivyo, ni moja kwa moja.

    1.


    1. Takwimu hii inaonyesha grafu ya mduara na kituo cha asili na radius 3.
      Kielelezo 10.1.65

    2. Takwimu hii inaonyesha grafu ya parabola kufungua juu na vertex saa (0k, 2).
      Kielelezo 10.1.66

    2.


    1. Takwimu hii inaonyesha ufunguzi wa parabola kwa haki na vertex saa (hasi 2, 0).
      Kielelezo 10.1.67

    2. Takwimu hii inaonyesha grafu ya polynomial yenye utaratibu usio wa kawaida, ili ianze katika roboduara ya tatu, huongezeka hadi asili na kisha inaendelea kuongezeka kwa njia ya roboduara ya kwanza.
      Kielelezo 10.1.68

    3.


    1. Takwimu hii inaonyesha grafu ya Curve inayoanza saa (hasi 6 hasi 2) huongezeka kwa asili na kisha inaendelea kuongezeka polepole kwa (6, 2).
      Kielelezo 10.1.69

    2. Takwimu hii inaonyesha parabola kufungua zaidi na kipeo katika (0, hasi 4).
      Kielelezo 10.1.70

    4.


    1. Takwimu hii inaonyesha sehemu ya mstari wa moja kwa moja inapungua kutoka (hasi 4, 6) hadi (2, 0), baada ya hapo huongezeka kutoka (2, 0) hadi (6, 4).
      Kielelezo 10.1.71

    2. Takwimu hii inaonyesha mduara na radius 4 na katikati ya asili.
      Kielelezo 10.1.72
    Jibu

    1.

    1. Si kazi
    2. Kazi; si moja kwa moja

    3.

    1. Kazi moja kwa moja
    2. Kazi; si moja kwa moja
    Zoezi\(\PageIndex{23}\) Determine Whether a Function is One-to-One

    Katika mazoezi yafuatayo, tafuta inverse ya kila kazi. Tambua kikoa na upeo wa kazi ya inverse.

    1. \(\{(2,1),(4,2),(6,3),(8,4)\}\)
    2. \(\{(6,2),(9,5),(12,8),(15,11)\}\)
    3. \(\{(0,-2),(1,3),(2,7),(3,12)\}\)
    4. \(\{(0,0),(1,1),(2,4),(3,9)\}\)
    5. \(\{(-2,-3),(-1,-1),(0,1),(1,3)\}\)
    6. \(\{(5,3),(4,2),(3,1),(2,0)\}\)
    Jibu

    1. \(\begin{array}{l}{\text { Inverse function: }\{(1,2),(2,4),(3,6),(4,8)\} . \text { Domain: }\{1,2,3,4\} . \text { Range: }} {\{2,4,6,8\} .}\end{array}\)

    3. \(\begin{array}{l}{\text { Inverse function: }\{(-2,0),(3,1),(7,2),(12,3)\} . \text { Domain: }\{-2,3,7,12\} \text { . }} {\text { Range: }\{0,1,2,3\}}\end{array}\)

    5. \(\begin{array}{l}{\text { Inverse function: }\{(-3,-2),(-1,-1),(1,0),(3,1)\} . \text { Domain: }} {\{-3,-1,1,3\} . \text { Range: }\{-2,-1,0,1\}}\end{array}\)

    Zoezi\(\PageIndex{24}\) Determine Whether a Function is One-to-One

    Katika mazoezi yafuatayo, grafu, kwenye mfumo huo wa kuratibu, inverse ya kazi moja kwa moja iliyoonyeshwa.


    1. Takwimu hii inaonyesha mfululizo wa makundi ya mstari kutoka (hasi 4, hasi 3) hadi (hasi 3, 0) kisha (hasi 1, 2) na kisha (3, 4).
      Kielelezo 10.1.73

    2. Takwimu hii inaonyesha mfululizo wa makundi ya mstari kutoka (hasi 4, hasi 4) hadi (hasi 3, 1) kisha (0, 2) na kisha (2, 4).
      Kielelezo 10.1.74

    3. Takwimu hii inaonyesha mfululizo wa makundi ya mstari kutoka (hasi 4, 4) hadi (0, 3) kisha (3, 2) na kisha (4, hasi 1).
      Kielelezo 10.1.75

    4. Takwimu hii inaonyesha mfululizo wa makundi ya mstari kutoka (hasi 4, hasi 4) hadi (hasi 1, hasi 3) kisha (0, 1), kisha (1, 3), na kisha (4, 4).
      Kielelezo 10.1.76
    Jibu

    1.

    Takwimu hii inaonyesha mfululizo wa makundi ya mstari kutoka (hasi 3, hasi 4) hadi (0, hasi 3) kisha (2, hasi 1), na kisha (4, 3).
    Kielelezo 10.1.77

    3.

    Takwimu hii inaonyesha mfululizo wa makundi ya mstari kutoka (hasi 1, 4) hadi (2, 3) kisha (3, 0), na kisha (4, hasi 4).
    Kielelezo 10.1.78
    Zoezi\(\PageIndex{25}\) Determine Whether the given functions are inverses

    Katika mazoezi yafuatayo, onyesha kama kazi zilizopewa ni inverses au si.

    1. \(f(x)=x+8\)na\(g(x)=x-8\)
    2. \(f(x)=x-9\)na\(g(x)=x+9\)
    3. \(f(x)=7 x\)na\(g(x)=\frac{x}{7}\)
    4. \(f(x)=\frac{x}{11}\)na\(g(x)=11 x\)
    5. \(f(x)=7 x+3\)na\(g(x)=\frac{x-3}{7}\)
    6. \(f(x)=5 x-4\)na\(g(x)=\frac{x-4}{5}\)
    7. \(f(x)=\sqrt{x+2}\)na\(g(x)=x^{2}-2\)
    8. \(f(x)=\sqrt[3]{x-4}\)na\(g(x)=x^{3}+4\)
    Jibu

    1. \(g(f(x))=x,\)na\(f(g(x))=x,\) hivyo wao ni inverses.

    3. \(g(f(x))=x,\)na\(f(g(x))=x,\) hivyo wao ni inverses.

    5. \(g(f(x))=x,\)na\(f(g(x))=x,\) hivyo wao ni inverses.

    7. \(g(f(x))=x,\)na\(f(g(x))=x,\) hivyo wao ni inverses (kwa nonnegative\(x )\)

    Zoezi\(\PageIndex{26}\) Determine the inverse of a function

    Katika mazoezi yafuatayo, tafuta inverse ya kila kazi.

    1. \(f(x)=x-12\)
    2. \(f(x)=x+17\)
    3. \(f(x)=9 x\)
    4. \(f(x)=8 x\)
    5. \(f(x)=\frac{x}{6}\)
    6. \(f(x)=\frac{x}{4}\)
    7. \(f(x)=6 x-7\)
    8. \(f(x)=7 x-1\)
    9. \(f(x)=-2 x+5\)
    10. \(f(x)=-5 x-4\)
    11. \(f(x)=x^{2}+6, x \geq 0\)
    12. \(f(x)=x^{2}-9, x \geq 0\)
    13. \(f(x)=x^{3}-4\)
    14. \(f(x)=x^{3}+6\)
    15. \(f(x)=\frac{1}{x+2}\)
    16. \(f(x)=\frac{1}{x-6}\)
    17. \(f(x)=\sqrt{x-2}, x \geq 2\)
    18. \(f(x)=\sqrt{x+8}, x \geq-8\)
    19. \(f(x)=\sqrt[3]{x-3}\)
    20. \(f(x)=\sqrt[3]{x+5}\)
    21. \(f(x)=\sqrt[4]{9 x-5}, x \geq \frac{5}{9}\)
    22. \(f(x)=\sqrt[4]{8 x-3}, x \geq \frac{3}{8}\)
    23. \(f(x)=\sqrt[5]{-3 x+5}\)
    24. \(f(x)=\sqrt[5]{-4 x-3}\)
    Jibu

    1. \(f^{-1}(x)=x+12\)

    3. \(f^{-1}(x)=\frac{x}{9}\)

    5. \(f^{-1}(x)=6 x\)

    7. \(f^{-1}(x)=\frac{x+7}{6}\)

    9. \(f^{-1}(x)=\frac{x-5}{-2}\)

    11. \(f^{-1}(x)=\sqrt{x-6}\)

    13. \(f^{-1}(x)=\sqrt[3]{x+4}\)

    15. \(f^{-1}(x)=\frac{1}{x}-2\)

    17. \(f^{-1}(x)=x^{2}+2, x \geq 0\)

    19. \(f^{-1}(x)=x^{3}+3\)

    21. \(f^{-1}(x)=\frac{x^{4}+5}{9}, x \geq 0\)

    23. \(f^{-1}(x)=\frac{x^{5}-5}{-3}\)

    Zoezi\(\PageIndex{27}\) Writing Exercises
    1. Eleza jinsi grafu ya inverse ya kazi inahusiana na grafu ya kazi.
    2. Eleza jinsi ya kupata inverse ya kazi kutoka equation yake. Tumia mfano ili kuonyesha hatua.
    Jibu

    1. Majibu yatatofautiana.

    Self Check

    Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

    Jedwali hili lina safu nne na nguzo nne. mstari wa kwanza, ambayo hutumika kama header, anasoma mimi canâ € |, Kwa ujasiri, Kwa msaada fulani, na Noâ €” Mimi donâ €™ t kupata hiyo. Safu ya kwanza chini ya mstari wa kichwa inasoma Tafuta na kutathmini kazi za vipande, onyesha kama kazi ni moja kwa moja, na upate inverse ya kazi. Wengine wa seli ni tupu.
    Kielelezo 10.1.79

    b Kama wengi wa hundi yako walikuwa:

    ... kwa ujasiri. Hongera! Umefanikiwa malengo katika sehemu hii. Fikiria ujuzi wa kujifunza uliyotumia ili uweze kuendelea kuitumia. Ulifanya nini ili uwe na ujasiri wa uwezo wako wa kufanya mambo haya? Kuwa maalum.

    ... kwa msaada fulani. Hii lazima kushughulikiwa haraka kwa sababu mada huna bwana kuwa mashimo katika barabara yako ya mafanikio. Katika hesabu kila mada hujenga juu ya kazi ya awali. Ni muhimu kuhakikisha kuwa na msingi imara kabla ya kuendelea. Nani unaweza kuomba msaada? Washiriki wenzako na mwalimu ni rasilimali nzuri. Je, kuna mahali kwenye chuo ambapo waalimu hisabati zinapatikana? Je, ujuzi wako wa kujifunza unaweza kuboreshwa?

    ... Hapana - siipati! Hii ni ishara ya onyo na haipaswi kupuuza. Unapaswa kupata msaada mara moja au utazidiwa haraka. Angalia mwalimu wako haraka iwezekanavyo kujadili hali yako. Pamoja unaweza kuja na mpango wa kupata msaada unayohitaji.