Matemàtiques - 1r BAT - Exercicis funció inversa
1-Donades `f(x)=sqrt(x-3)`, `g(x)=3x+2` i `h(x)=(2x-1)/(x+1)`.
a) Calcula les seves funcions inverses.
- `y=sqrt(x-3) => x=sqrt(y-3) => x^2=y-3 => y=x^2+3 => f^(-1)(x)=x^2+3`
- `y=3x+2 => x=3y+2 => y=(x-2)/3 => g^(-1)(x)=(x-2)/3`
- `y=(2x-1)/(x+1) => x=(2y-1)/(y+1) => x·(y+1)=2y-1`
`=xy+x=2y-1 => xy-2y = -x-1 => (x-2)y=-x-1 =>`
`y=(-x-1)/(x-2) => h^(-1)(x)=(-x-1)/(x-2)`
b) Comprova que ho són.
- `f^(-1)(f(x))=sqrt((x^2+3)-3)=sqrt(x^2+3-3)=sqrt(x^2)=x`
- `g^(-1)(g(x))=3(x-2)/3+2=x-2+2=x`
- `h^(-1)(h(x))=(-((2x-1)/(x+1))-1)/(((2x-1)/(x+1))-2)=((-2x+1)/(x+1)-(x+1)/(x+1))/((2x-1)/(x+1)-(2(x+1))/(x+1))=(((-2x+1)-(x+1))/(x+1))/(((2x-1)-(2x+2))/(x+1))=(-3x)/(-3)=x`
2-Donada la funció `f(x)=x/(x-1)`.
a) Calcula `f(f(x))`.
`f(f(x))=(x/(x-1))/((x/(x-1))-1)=(x/(x-1))/(x/(x-1)-(x-1)/(x-1))=(x/(x-1))/((x-(x-1))/(x-1))=(x/(x-1))/(1/(x-1))=x/1=x`
b) Quina propietat compleix `f(x)`?
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