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Math Help - Composition of Two Functions

  1. #1
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    Composition of Two Functions

    Let f(x) = x + 4 and h(x) = 4x - 1. Find a function g such that g \circ f = h.


    g(f(x)) = h

    g(f) = h

    g(x + 4) = 4x - 1

    g = \frac {4x - 1}{x + 4}


    And textbook answer is g(x) = 4x - 17

    I don't know how to get the textbook answer.......?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Macleef View Post
    Let f(x) = x + 4 and h(x) = 4x - 1. Find a function g such that g \circ f = h.


    g(f(x)) = h

    g(f) = h

    g(x + 4) = 4x - 1

    \color{red}g = \frac {4x - 1}{x + 4}


    And textbook answer is g(x) = 4x - 17
    the line in red is just wrong. g(x + 4) is NOT a product. you do not think of it as g times (x + 4). it is g of (x + 4), it is function notation, you cannot manipulate it as you did.

    now when you have g(x + 4), it means you took some function g(x) and shifted it 4 units to the left. so given g(x + 4), we must replace x with x - 4, that way, we have g(x - 4 + 4) = g(x), what we did here was shift the function back to the right. thus we know that to get g from h, we must replace the x in h(x) with x - 4 (essentially what we are doing is shifting the h(x) function to the right to match up with the g(x)).

    so, we have:

    g(x + 4) = h(x)

    replace x everywhere with x - 4, we get:

    g(x) = h(x - 4)

    => g(x) = 4(x - 4) - 1

    => g(x) = 4x - 17
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