precal

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For the given functions f, g, and h, find fogoh and state the exact domain of fogoh. Please show all of your work.
f(x)=lnx
g(x)=|x|-2
h(x)=4x-7

  • precal -

    domains:

    f: x>0
    g: all reals
    h: all reals

    f(x) = lnx
    f(g(h)) = ln(g(h))
    = ln(|h|-2)
    = ln(|4x-7|-2)

    ln(u) is defined only for u>0
    so, we need |4x-7|-2 > 0
    |4x-7| > 2
    so, if 4x-7>=0 (or x>=7/4), 4x-7 > 2 ==> x > 9/4

    or, if 4x-7 < 0 (or x<7/4), -(4x-7) > 2 ==> x < 5/4

    so, domain is (-oo,5/4) U [9/4,oo)

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