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Perhatikan grafik berikut. Daerah hasil grafik fungsi y=f(x) adalah ...

Pertanyaan

Perhatikan grafik berikut.

Daerah hasil grafik fungsi begin mathsize 14px style y equals straight f open parentheses x close parentheses end style adalah ...

  1. begin mathsize 14px style open curly brackets straight y vertical line minus 7 less than straight y less or equal than 3 comma space straight y element of straight R close curly brackets end style

  2. begin mathsize 14px style left curly bracket straight y vertical line minus 7 less or equal than straight y less than 3 comma space straight y element of straight R right curly bracket end style

  3. begin mathsize 14px style left curly bracket straight y vertical line minus 4 less than straight y less or equal than 2 comma space straight y element of straight R right curly bracket end style

  4. begin mathsize 14px style left curly bracket straight y vertical line minus 4 less or equal than straight y less than 2 comma space straight y element of straight R right curly bracket end style  

  5. begin mathsize 14px style left curly bracket straight y vertical line minus 4 less or equal than straight y less than 7 comma space straight y element of straight R right curly bracket end style

Pembahasan Video:

Pembahasan Soal:

Dari gambar di atas, terlihat bahwa daerah hasil terdapat di antara begin mathsize 14px style negative 4 end style (bulatan penuh) dan size 14px 2 (bulatan kosong), sehingga dapat ditulis negative 4 less or equal than straight y less than 2.

Jadi, jawaban yang tepat adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 07 Oktober 2021

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Pertanyaan yang serupa

Diketahui fungsi f(x)=xx−3​, x=0 dan g(x)=x2−9​. Tentukan rumus fungsi berikut apabila terdefinisi dan tentukan daerah asal dan daerah hasilnya. a)   f+g

Pembahasan Soal:

Diketahui fungsi  dan , maka fungsi open parentheses f plus g close parentheses open parentheses x close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f plus g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses x close parentheses plus g open parentheses x close parentheses end cell row blank equals cell fraction numerator x minus 3 over denominator x end fraction plus square root of x squared minus 9 end root end cell end table

Fungsi f open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x end fraction terdefinisi saat x not equal to 0, maka D subscript f equals open curly brackets x vertical line space x not equal to 0 comma space x element of straight real numbers close curly brackets. Kemudian, fungsi g open parentheses x close parentheses equals square root of x squared minus 9 end root terdefinisi saat x squared minus 9 greater or equal than 0. Nilai x yang memenuhi saat x squared minus 9 greater or equal than 0 sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared minus 9 end cell greater or equal than 0 row cell open parentheses x minus 3 close parentheses open parentheses x plus 3 close parentheses end cell greater or equal than 0 end table

Berdasarkan uraian di atas, x squared minus 9 greater or equal than 0 saat x minus 3 greater or equal than 0 dan x plus 3 greater or equal than 0 atau x minus 3 less or equal than 0 atau x plus 3 less or equal than 0. Saat x minus 3 greater or equal than 0 dan x plus 3 greater or equal than 0, nilai x yang memenuhi sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 3 end cell greater or equal than 0 row x greater or equal than 3 end table


table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 3 end cell greater or equal than 0 row x greater or equal than cell negative 3 end cell end table

x greater or equal than 3 dan x greater or equal than negative 3, maka nilai x yang memenuhi keduanya adalah x greater or equal than 3. Lalu, saat x minus 3 less or equal than 0 atau x plus 3 less or equal than 0, nilai x yang memenuhi sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 3 end cell less or equal than 0 row x less or equal than 3 end table


table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 3 end cell less or equal than 0 row x less or equal than cell negative 3 end cell end table

x less or equal than 3 dan x less or equal than negative 3, maka nilai x yang memenuhi keduanya adalah x less or equal than negative 3.

Berdasarkan uraian di atas, x squared minus 9 greater or equal than 0 saat x greater or equal than 3 atau x less or equal than negative 3, maka D subscript g equals open curly brackets x vertical line space x less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets.

Berdasarkan daerah asal fungsi f dan fungsi g di atas, maka diperoleh D subscript open parentheses f plus g close parentheses end subscript atau daerah asal fungsi open parentheses f plus g close parentheses sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell D subscript open parentheses f plus g close parentheses end subscript end cell equals cell D subscript f intersection D subscript g end cell row blank equals cell open curly brackets x vertical line space x not equal to 0 comma space x element of straight real numbers close curly brackets intersection open curly brackets x vertical line x less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets end cell row blank equals cell open curly brackets x vertical line space x less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets end cell end table


Selanjutnya, daerah hasil fungsi open parentheses f plus g close parentheses open parentheses x close parentheses di atas adalah open curly brackets y vertical line space y greater or equal than 0 comma space y element of straight real numbers close curly brackets

Dengan demikian, fungsi open parentheses f plus g close parentheses adalah open parentheses f plus g close parentheses open parentheses x close parentheses equals fraction numerator x minus 3 over denominator x end fraction plus square root of x squared minus 9 end root, daerah asalnya adalah D subscript open parentheses f plus g close parentheses end subscript equals open curly brackets x vertical line space x less or equal than negative 3 space atau space x greater or equal than 3 comma space x element of straight real numbers close curly brackets, dan daerah hasilnya adalahR subscript open parentheses f plus g close parentheses end subscript equals open curly brackets y vertical line space y greater or equal than 0 comma space y element of straight real numbers close curly brackets.

1

Roboguru

Diketahui fungsi f(x)=x2+4x+3 dengan daerah asal {x∣−3≤x≤0,x∈R}. Daerah hasil fungsi f adalah....

Pembahasan Soal:

Perhatikan penghitungan berikut!space 

 f open parentheses x close parentheses equals x squared plus 4 x plus 3 diubah ke bentuk kuadrat sempurna:

f open parentheses x close parentheses equals open parentheses x plus open parentheses 1 half cross times 4 close parentheses close parentheses squared minus open parentheses 1 half cross times 4 close parentheses squared plus 3 f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses squared minus open parentheses 2 close parentheses squared plus 3 f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses squared minus 1

Menentukan daerah hasil:

table attributes columnalign right center left columnspacing 0px end attributes row cell negative 3 end cell less or equal than cell x less or equal than 0 end cell row cell negative 3 plus 2 end cell less or equal than cell x plus 2 less or equal than 0 plus 2 end cell row cell negative 1 end cell less or equal than cell x plus 2 less or equal than 2 end cell row 0 less or equal than cell open parentheses x plus 2 close parentheses squared less or equal than 2 squared space end cell row 0 less or equal than cell open parentheses x plus 2 close parentheses squared less or equal than 4 end cell row cell 0 minus 1 end cell less or equal than cell open parentheses x plus 2 close parentheses squared minus 1 less or equal than 4 minus 1 end cell row cell negative 1 end cell less or equal than cell open parentheses x plus 2 close parentheses squared minus 1 less or equal than 3 end cell row cell negative 1 end cell less or equal than cell f open parentheses x close parentheses less or equal than 3 end cell row cell negative 1 end cell less or equal than cell y less or equal than 3 end cell end table

Daerah space hasil equals open curly brackets y space left enclose space minus 1 less or equal than y less or equal than 3 comma space y element of straight real numbers end enclose close curly brackets.

Oleh karena itu, jawaban yang benar adalah B. space 

0

Roboguru

Tentukanlah daerah asal dan daerah hasil fungsi berikut.  d. f(x)=x(x−5)2​

Pembahasan Soal:

f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x left parenthesis x minus 5 right parenthesis end fraction

Daerah asal 

Karena fungsi memuat bentuk pecahan, maka agar terdefinisi penyebut undefined

Sehingga

x left parenthesis x minus 5 right parenthesis equals 0 x equals 0 space atau space straight x minus 5 equals 0 x italic equals italic 0 space atau space x italic equals italic 5

Daerah asal fungsi    undefined

 

Daerah hasil 

Berdasarkan grafik f left parenthesis x right parenthesis equals fraction numerator 2 over denominator x left parenthesis x minus 5 right parenthesis end fraction, dapat dilihat bahwa asimtot datarnya = 0, maka daerah hasilnya not equal to 0

Sehingga daerah hasil fungsi  left curly bracket f left parenthesis x right parenthesis vertical line f left parenthesis x right parenthesis not equal to 0 comma space f left parenthesis x right parenthesis element of straight real numbers right curly bracket

Dengan demikian, Daerah asal fungsi undefined dan daerah hasil fungsi  left curly bracket f left parenthesis x right parenthesis vertical line f left parenthesis x right parenthesis not equal to 0 comma space f left parenthesis x right parenthesis element of straight real numbers right curly bracket.

0

Roboguru

Jika fungsi g(x)=2x2−8x−7 maka daerah hasilnya adalah ...

Pembahasan Soal:

Diketahui:

Fungsi kuadrat g open parentheses x close parentheses equals 2 x squared minus 8 x minus 7.

Bentuk umum fungsi kuadrat adalah g open parentheses x close parentheses equals a x plus b x plus c.

Karena a greater than 0, maka grafik fungsi terbuka keatas, sehingga range fungsinya, yaitu:

y greater or equal than g open parentheses negative fraction numerator b over denominator 2 a end fraction close parentheses

Diperoleh:

y greater or equal than g open parentheses negative fraction numerator b over denominator 2 a end fraction close parentheses y greater or equal than g open parentheses negative fraction numerator negative 8 over denominator 2 open parentheses 2 close parentheses end fraction close parentheses y greater or equal than g open parentheses 8 over 4 close parentheses y greater or equal than g open parentheses 2 close parentheses

Kemudian tentukan nilai fungsi g open parentheses x close parentheses equals 2 x squared minus 8 x minus 7.

y greater or equal than g open parentheses x close parentheses y greater or equal than g open parentheses 2 close parentheses y greater or equal than 2 open parentheses 2 close parentheses squared minus 8 open parentheses 2 close parentheses minus 7 y greater or equal than 8 minus 16 minus 7 y greater or equal than negative 15

Maka, daerah hasilnya adalah R subscript g equals open curly brackets right enclose y space y greater or equal than negative 15 comma space y element of straight real numbers close curly brackets.

0

Roboguru

Daerah hasil fungsi f(x)=x2+4x−2 untuk daerah asal {x∣−2≤x≤4,x∈R} dan y=f(x) adalah...

Pembahasan Soal:

Menentukan daerah hasil (range) dari suatu fungsi

Fungsi

f open parentheses x close parentheses equals x squared plus 4 x minus 2 

Domain (daerah asal)

table attributes columnalign right center left columnspacing 0px end attributes row D equals cell open curly brackets x vertical line minus 2 less or equal than x less or equal than 4 comma space x element of R close curly brackets end cell row blank equals cell open curly brackets negative 2 comma negative 1 comma 0 comma 1 comma 2 comma 3 comma 4 close curly brackets end cell end table 

Menentukan kodomain (daerah kawan)

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell x squared plus 4 x minus 2 end cell row cell f open parentheses negative 2 close parentheses end cell equals cell open parentheses negative 2 close parentheses squared plus 4 open parentheses negative 2 close parentheses minus 2 end cell row blank equals cell 4 minus 8 minus 2 end cell row blank equals cell negative 6 end cell row cell f open parentheses negative 1 close parentheses end cell equals cell open parentheses negative 1 close parentheses squared plus 4 open parentheses negative 1 close parentheses minus 2 end cell row blank equals cell 1 minus 4 minus 2 end cell row blank equals cell negative 2 end cell row cell f open parentheses 0 close parentheses end cell equals cell open parentheses 0 close parentheses squared plus 4 open parentheses 0 close parentheses minus 2 end cell row blank equals cell 0 minus 0 minus 2 end cell row blank equals cell negative 2 end cell row cell f open parentheses 1 close parentheses end cell equals cell open parentheses 1 close parentheses squared plus 4 open parentheses 1 close parentheses minus 2 end cell row blank equals cell 1 plus 4 minus 2 end cell row blank equals 3 row cell f open parentheses 2 close parentheses end cell equals cell open parentheses 2 close parentheses squared plus 4 open parentheses 2 close parentheses minus 2 end cell row blank equals cell 4 plus 8 minus 2 end cell row blank equals 10 row cell f open parentheses 3 close parentheses end cell equals cell open parentheses 3 close parentheses squared plus 4 open parentheses 3 close parentheses minus 2 end cell row blank equals cell 9 plus 12 minus 2 end cell row blank equals 19 row cell f open parentheses 4 close parentheses end cell equals cell open parentheses 4 close parentheses squared plus 4 open parentheses 4 close parentheses minus 2 end cell row blank equals cell 16 plus 16 minus 2 end cell row blank equals 30 end table 

K equals open curly brackets negative 6 comma negative 2 comma negative 2 comma 3 comma 10 comma 19 comma 30 close curly brackets 

Menentukan range (daerah hasil)

R equals open curly brackets open parentheses negative 2 comma negative 6 close parentheses semicolon open parentheses negative 1 comma negative 2 close parentheses semicolon open parentheses 0 comma negative 2 close parentheses semicolon open parentheses 1 comma 3 close parentheses semicolon open parentheses 2 comma 10 close parentheses semicolon open parentheses 3 comma 19 close parentheses semicolon open parentheses 4 comma 30 close parentheses close curly brackets 

Jadi, daerah hasil fungsi di atas adalah

R equals open curly brackets open parentheses negative 2 comma negative 6 close parentheses semicolon open parentheses negative 1 comma negative 2 close parentheses semicolon open parentheses 0 comma negative 2 close parentheses semicolon open parentheses 1 comma 3 close parentheses semicolon open parentheses 2 comma 10 close parentheses semicolon open parentheses 3 comma 19 close parentheses semicolon open parentheses 4 comma 30 close parentheses close curly brackets.

0

Roboguru

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