Lukislah daerah himpunan penyelesaian (DHP) dari masing-masing sistem pertidaksamaan di bawah ini. f. {y≥x2−2y<−x2

Pertanyaan

Lukislah daerah himpunan penyelesaian (DHP) dari masing-masing sistem pertidaksamaan di bawah ini.

f. $\left\{\begin{array}{l}y\geq x^2-2\\y<-x^2\end{array}\right.$

Pembahasan:

Cari DHP dari $y greater or equal than x squared minus 2$ terlebih dahulu

Kurva pembatas dari $y greater or equal than x squared minus 2$ adalah $y equals x squared minus 2$

Berdasarkan bentuk umum fungsi kuadrat $y equals a x squared plus b x plus c$ diketahui

$a equals 1 b equals 0 c equals negative 2$

Cari titik puncak pada bentuk fungsi kuadrat $y equals a x squared plus b x plus c$ seperti berikut

$table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x subscript p comma space y subscript p right parenthesis end cell equals cell left parenthesis negative fraction numerator b over denominator 2 a end fraction comma space minus fraction numerator b squared minus 4 a c over denominator 4 a end fraction right parenthesis end cell row blank equals cell left parenthesis negative fraction numerator 0 over denominator 2 left parenthesis 1 right parenthesis end fraction comma space minus fraction numerator 0 squared minus 4 left parenthesis 1 right parenthesis left parenthesis negative 2 right parenthesis over denominator 4 left parenthesis 1 right parenthesis end fraction right parenthesis end cell row blank equals cell left parenthesis 0 comma space minus 8 over 4 right parenthesis end cell row blank equals cell left parenthesis 0 comma space minus 2 right parenthesis end cell end table$

Cari titik-titik pembentuk kurva tersebut

Untuk $x equals 2$

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared minus 2 end cell row y equals cell 2 squared minus 2 end cell row y equals cell 4 minus 2 end cell row y equals 2 row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis 2 comma space 2 right parenthesis end cell end table$

Untuk $x equals negative 2$

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell x squared minus 4 end cell row y equals cell left parenthesis negative 2 right parenthesis squared minus 2 end cell row y equals cell 4 minus 2 end cell row y equals 2 row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis negative 2 comma space 2 right parenthesis end cell end table$

Untuk mengetahui DHP lakukan uji titik

Untuk $left parenthesis 0 comma space 1 right parenthesis$

$y greater or equal than x squared minus 2 1 greater or equal than 0 squared minus 2 1 greater or equal than negative 2 space left parenthesis benar right parenthesis$

Karena titik $left parenthesis 0 comma space 1 right parenthesis$ benar maka daerah yang terdapat titik $left parenthesis 0 comma space 1 right parenthesis$ merupakan DHP dari $y greater or equal than x squared minus 2$

Selanjutnya cari DHP dari $y less than negative x squared$

Kurva pembatas dari $y less than negative x squared$ adalah $y equals negative x squared$

Berdasarkan bentuk umum fungsi kuadrat $y equals a x squared plus b x plus c$ diketahui

$a equals negative 1 b equals 0 c equals 0$

Cari titik puncak pada bentuk fungsi kuadrat $y equals a x squared plus b x plus c$ seperti berikut

$table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x subscript p comma space y subscript p right parenthesis end cell equals cell left parenthesis negative fraction numerator b over denominator 2 a end fraction comma space minus fraction numerator b squared minus 4 a c over denominator 4 a end fraction right parenthesis end cell row blank equals cell left parenthesis negative fraction numerator 0 over denominator 2 left parenthesis negative 1 right parenthesis end fraction comma space minus fraction numerator 0 squared minus 4 left parenthesis negative 1 right parenthesis left parenthesis 0 right parenthesis over denominator 4 left parenthesis negative 1 right parenthesis end fraction right parenthesis end cell row blank equals cell left parenthesis 0 comma space minus fraction numerator 0 over denominator negative 4 end fraction right parenthesis end cell row blank equals cell left parenthesis 0 comma space 0 right parenthesis end cell end table$

Cari titik-titik pembentuk kurva tersebut

Untuk $x equals 2$

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative x squared end cell row y equals cell negative left parenthesis 2 right parenthesis squared space end cell row y equals cell negative 4 end cell row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis 2 comma space minus 4 right parenthesis end cell end table$

Untuk $x equals negative 2$

$table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative x squared end cell row y equals cell negative left parenthesis negative 2 right parenthesis squared space end cell row y equals cell negative 4 end cell row cell left parenthesis x comma space y right parenthesis end cell equals cell left parenthesis negative 2 comma space minus 4 right parenthesis end cell end table$

Untuk mengetahui DHP lakukan uji titik

Untuk $left parenthesis 0 comma space 1 right parenthesis$

$table attributes columnalign right center left columnspacing 0px end attributes row y less than cell negative x squared end cell row 1 less than cell negative 0 squared end cell row 1 less than cell 0 space left parenthesis salah right parenthesis end cell end table$

Karena titik $left parenthesis 0 comma space 1 right parenthesis$ salah maka daerah yang terdapat titik $left parenthesis 0 comma space 1 right parenthesis$ bukan merupakan DHP dari $y less than negative x squared$

DHP dari $open curly brackets table attributes columnalign left end attributes row cell y greater or equal than x squared minus 2 end cell row cell y less than negative x squared end cell end table close$ merupakan irisan DHP dari $y greater or equal than x squared minus 2$ dan DHP $y less than negative x squared$

Jadi, DHP dari $open curly brackets table attributes columnalign left end attributes row cell y greater or equal than x squared minus 2 end cell row cell y less than negative x squared end cell end table close$ yaitu