พีชคณิตของฟังก์ชัน

[Rovers]

1. กำหนด f(x) = x+5 , (gof)(x) = x2 - 25 จงหา (f+g)(x) , (f-g)(x) , (f[:dot]g)(x) , (f/g)(x)


(gof)(x) = g(f(x)) = g(x+5)
g(x+5) = x2 - 25 = (x+5)(x-5) = (x+5)(x+5-10)
g(x) = x(x-10) = x2 - 10x

1.1 (f+g)(x)
= f(x) + g(x)
= (x+5) + (x2-10x)
= x2 - 9x +5
1.2 (f-g)(x)
= f(x) - g(x)
= (x+5) - (x2-10x)
= 11x + 5 - x2
1.3 (f[:dot]g)(x)
= f(x)g(x)
= (x+5)(x2-10x)
= x3 - 5x2 - 50x
1.4 (f/g)(x)
= f(x)/g(x)
= (x+5) / (x2-10x) , x [:notequal] 0,10


2. กำหนด f(2x-3) = 3x-2 และ (f+g)(x) = x2+x-3 จงหา (fog)(x) , (gof)(x), (f-g)(x) , (f[:dot]g)(x) , (f/g)(x) และ (g/f)(x)



f(2x-3) = 3x-2 = (3(2x-3)+5)/2
f(x) = (3x+5)/2
(f+g)(x) = f(x)+g(x) = x2+x-3
g(x) = x2+x-3 - ((3x+5)/2) = (2x2-x-11)/2

[Rovers]

3. กำหนด f-1(x+1) = 2x+3 และ (fog)(x-1) = 5x+1
จงหา (f+g)(1) , (f+g)-1(1) , ((f[:dot]g)of-1)(1) , (f/g + f-1)(3)



f-1(x+1) = 2x+3 = 2(x+1)+1
f-1(x) = 2x+1
f(x) = (x-1)/2

(fog)(x-1) = f(g(x-1)) = (g(x-1)-1)/2 = 5x+1
g(x-1) = 10x+3 = 10(x-1) +13
g(x) = 10x+13

(f+g)(x) = f(x)+g(x) = (21x+25)/2
[:therefore] (f+g)-1(x) = (2x-25)/21



4. กำหนด f(x) = 3x+5 , g(x) = 2x+1 จงหา
4.1 (f-1+g-1)(5)
4.2 (f-1-g-1)(10)
4.3 (f-1[:dot]g-1)(-5)
4.4 (f-1/g-1)(-10)



f(x) = 3x+5
f-1(x) = (x-5)/3
g(x-1) = 2x+1 = 2(x-1)+3
g(x) = 2x+3
g-1(x) = (x-3)/2