? asked in 科學及數學數學 · 1 decade ago

Maths For Finance

Let f (x) = x + 2x2 − 1 and g (x) = √6 − 2x.

(a) Find the domains of f (x) and g (x) .

(b) Find the function f ◦ g (x) and g ◦ f (x) .

(c) Find the inverse of g (x) .

(d) Show that g^−1 (g (x)) = x.

1 Answer

Rating
  • 學問
    Lv 4
    1 decade ago
    Favorite Answer

    (a) domain for f(x) is all real numbers,

    domain for g(x) is x 小於或等於 3, 因為 6 - 2 x不能為負數。

    (b) f(g(x)) = ( sqrt( 6-2x ) )+2( sqrt( 6-2x ) )^2-1

    = ( sqrt( 6-2x ) )+2( 6-2x )-1

    = sqrt( 6-2x ) + 11 - 4x

    g(f(x)) = √[ 6 − 2 ( x + 2x^2 − 1 ) ]

    = √[ 6 − 2 x + 4x^2 − 2 ) ]

    = √[ 4 − 2 x + 4x^2 ) ]

    (c) Let y = sqrt ( 6 - 2x )

    y^2 = 6 - 2x

    2x = 6 -y^2

    x = 3 - 1/2 * y^2

    g^-1(x) = 3 - 1/2 * x^2

    (d) g^-1(g(x))

    = 3 - 1/2 * g(x)^2

    = 3 - 1/2 * sqrt( 6 - 2x )^2

    = 3 - 1/2 *( 6 - 2x )

    = 3 - ( 3 - x )

    = 3 - 3 + x

    = x

    P.S.

    1. domain 指一個函數的定義域,

    入domain外的數字入個function, 就無意思喇!(例如 a part 的開方負數)

    2. inverse 係指函數的反函數,係將函數相反的過程。

    中一二常常做的主項轉換,就係搵反函數的最佳例子!

    Source(s): 唔明再問我la^^
    • Commenter avatarLogin to reply the answers
Still have questions? Get your answers by asking now.