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March 29, 2017

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1. Find the domain of the composite function f o g. f(x)= 2/x-3; g(x)= 7/x

2. Solve the following exponential equation. Exact answers only. 2^1-9x = e^2x

  • College Algebra - ,

    f(g(x)) = 2/(g(x)-3) = 2/(7/x)-3) = 2x/(7-3x)

    domain would be all reals except x = 7/3, but we have to recall that g(x) is not defined for x=0.

    So the domain of fog is all reals except 0, 7/3


    2^(1-9x) = e^2x

    take log of both sides, recalling that ln(a^b) = b*lna:

    (1-9x)ln2 = 2x
    ln2 - 9 ln2 * x = 2x
    2x + 9 ln2*x = ln2
    x = ln2/(2+9 ln2)
    or, if you want to be sneaky,

    ln2/(ln(e^2) + ln(2^9))
    = ln2/ln(512e^2)
    = log512e^22

  • College Algebra - ,

    1.
    fog(x) = 2 /[(7/x) - 3]
    simplifying,
    2x/(7 - 3x)
    Domain: all real numbers except 3, 0 and 7/3

    2.
    2^(1-9x) = e^(2x)
    get ln of both sides:
    ln 2^(1-9x) = ln e^(2x)
    (1-9x)*(ln 2) = 2x
    (1-9x)/(2x) = 1/(ln 2)
    1/(2x) - 9/2 = 1/(ln 2)
    1/(2x) = 1/(ln 2) + 9/2
    2x = 1/[1/(ln 2) + 4.5]
    x = 1/[2(1/ln 2 + 4.5)]
    x = 1/[2/(ln 2) + 9]
    x = 1 / [2/(ln 2) + 9(ln 2) / (ln 2)]
    x = 1 / [2 + 9(ln 2)] / ln 2
    finally,
    x = (ln 2)/(2 + (9 ln 2))

    hope this helps~ :)

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