Math

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how would i go about finding the domain of the function
f(x)=(3x+1)/(sqrt(x^2+x-2))
and both the domain and range of
g(x)=(5x-3)/(2x+1)

  • Math -

    Well, the domain would be all x except where the denominator is zero or the sqrt is of a negative number so we need to find where the denominator is zero and where x^2 + x - 2 is negative.
    let's look at that denominator function inside the radical sign:
    d = x^2+x -2
    It is a parabola that opens up
    first find the zeros
    0 = (x-1)(x+2)
    so it is zero at x = -2 and x =+1
    those points must not be in the domain but also all the points between them are out because the d function dips below zero between -2 and +1
    so our domain is -2 > x > +1
    -----------------------------
    Now for the second one the denominator is zero for x = -1/2 and all real x except x = -1/2 is the domain
    To find the range, sketch the function
    for x << 0, g(x)--> +5/2
    for x >> 0, g(x)--> +5/2
    for x = 0, g(x) = -3
    now look at where the numerator is zero
    like
    x = .6, g = 0
    now look at points close to x = -.5 where g(x) gets huge
    like
    x = -.6, g = 30
    x = -.4, g = -25
    that tells you what happens each side of the singularity at x = -.5
    We see that g(x) goes to +oo as x approaches -.5 from the left and g(x) goes to -oo as x approaches -.5 from the right.
    Therefore the range of g is from -oo to +oo

  • Math -

    i understood the whole first question and then i understood the domain of the second question. i did get confused though when it came to finding the range because the teacher gave us the answers w/o explanations and it said that the range was all numbers except for 5/2. i'm not sure how she got that.

    thanks for all your help!

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