tolong dong kak yang bisa​

Berikut ini adalah pertanyaan dari buried6feetunder pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Tolong dong kak yang bisa​
tolong dong kak yang bisa​

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

1. f(x) = x²

g(x) = 3x

h(x) = 3x² + 1

a) ( h o g o f )(x) = h o g [ f(x) ]

= h o g(x²)

= h [ 3(x²) ]

= h(3x²)

= 3(3x²) + 1

= 9x² + 1

( f o g o h )(x) = f o g [ h(x) ]

= f o g( 3x² + 1 )

= f [ 3( 3x² + 1 ) ]

= f( 9x² + 3 )

= ( 9x² + 3 )²

= 81x⁴ + 27x² + 27x² + 9

= 81x⁴ + 54x² + 9

b) ( g o h o f )(x) = g o h [ f(x) ]

= g o h(x²)

= g [ 3(x²)² + 1 ]

= g( 3x⁴ + 1 )

= 3( 3x⁴ + 1 )

= 9x⁴ + 3

( g o h o f )(-2) = 9(-2)⁴ + 3

= 9.16 + 3

= 144 + 3 = 147

( g o f o h )(x) = g o f [ h(x) ]

= g o f( 3x² + 1 )

= g [ ( 3x² + 1 )² ]

= g( 9x⁴ + 6x² + 1 )

= 3( 9x⁴ + 6x² + 1 )

= 27x⁴ + 18x² + 3

( g o f o h )(-2) = 27(-2)⁴ + 18(-2)² + 3

= 27.16 + 18.4 + 3

= 432 + 72 + 3

= 507

2. f(x) = 2x - 1

g(x) = 3x

h(x) = x² + 1

a) ( f o h o g )(x) = f o h [ g(x) ]

= f o h(3x)

= f [ (3x)² + 1 ]

= f( 9x² + 1 )

= 2( 9x² + 1 ) - 1

= 18x² + 2 - 1

= 18x² + 1

( f o h o g )(2) = 18.2² + 1

= 18.4 + 1

= 72 + 1 = 73

( h o g o f )(x) = h o g [ f(x) ]

= h o g( 2x - 1 )

= h [ 3( 2x - 1 ) ]

= h( 6x - 3 )

= ( 6x - 3 )² + 1

= 36x² - 36x + 9 + 1

= 36x² - 36x + 10

( h o g o f )(2) = 36.2² - 36.2 + 10

= 36.4 - 72 + 10

= 144 - 62 = 82

b) ( h o g o f )( x-1 )

= 36( x + 1 )² - 36( x + 1 ) + 10

= 36( x² + 2x + 1 ) - 36x - 36 + 10

= 36x² + 72x + 36 - 36x - 26

= 36x² + 36x +10

( f o g o h )(x) = f o g [ h(x) ]

= f o g( x² + 1 )

= f [ 3( x² + 1 ) ]

= f( 3x² + 3 )

= 2( 3x² + 3 ) - 1

= 6x² + 6 - 1

= 6x² + 5

( f o g o h )( x - 1 ) = 6( x - 1 )² + 5

= 6( x² - 2x + 1 ) + 5

= 6x² - 12x + 6 + 5

= 6x² - 12x + 11

3. a) f(x) = y

3x - 2 = y

3x = y + 2

x = \frac{y + 2}{3} \\ f {}^{ - 1} (x) = \frac{x + 2}{3}

g(x) = y

 \frac{5x + 2}{3x - 2} = y

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

5x + 2 = 3yx - 2y

5x - 3yx = -2y - 2

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

x = \frac{ - 2y - 2}{5 - 3y} \\ f {}^{ - 1} (x) = \frac{ - 2x - 2}{5 - 3x}

b) ( f⁻¹ o g⁻¹ ) = f⁻¹ [ g⁻¹(x) ]

 = f {}^{ - 1} ( \frac{ - 2x - 2}{5 - 3x} ) \\ = \frac{ \frac{ - 2x - 2}{5 - 3x} + 2 }{3} \\ = \frac{ \frac{ - 2x - 2 + 2(5 - 3x)}{5 - 3x} }{3} \\ = \frac{ \frac{ - 2x - 2 + 10 - 6x}{5 - 3x} }{3} \\ = \frac{ \frac{ - 8x + 8}{5 - 3x} }{3} \\ = \frac{ - 8x + 8}{5 - 3x} . \frac{1}{3} = \frac{ - 8x + 8}{15 - 9x}

( g o f )(x) = g [ f(x) ]

= g( 3x - 2 )

 = \frac{5(3x - 2) + 2}{3(3x - 2) - 2} \\ = \frac{15x - 10 + 2}{9x - 6 - 2} \\ = \frac{15x - 8}{9x - 8}

( g o f )(x) = y

 \frac{15x - 8}{9x - 8} = y

15x - 8 = y( 9x - 8 )

15x - 8 = 9yx - 8y

15x - 9yx = -8x + 8

( 15 - 9y ) x = -8x + 8

x = \frac{ - 8y + 8}{15 - 9y} \\ f {}^{ - 1} (x) = \frac{ - 8x + 8}{15 - 9x}

4. ( g o f )(x) = g [ f(x) ]

4x² + 8x - 3 = g( 2x + 4 )

Misal : 2x + 4 = R

2x = R - 4

x = ( R - 4 )/2

4[ ( R - 4 )/2 ]² + 8[ ( R - 4 )/2 ] - 3 = g(R)

4[ ( R² - 8R + 16 )/4 ] + 4( R - 4 ) - 3 = g(R)

R² - 8R + 16 + 4R - 16 - 3 = g(R)

R² - 4R - 3 = g(R)

g(x) = x² - 4x - 3

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Last Update: Sat, 07 May 22