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Does the function y = 2^x ever cross the x-axis? Explain why or why not

  • algebra -

    The point where the function crosses the x-axis is call the x-intercept or
    solution. Y equals zero at the point where the graph croses the x-axis. If
    Y never goes to zero, the function does not cross or touch the x-axis.

    It can be proven that the graph of the
    given Eq does not cross or touch the x-axis for any real value of x:

    Let x = 0,
    Y = 2^x = 2^0 = 1.
    So when x is positive, the minimum value of Y is 1.

    Let x = -10.
    Y = 2^-10 = 1/2^10 = 0.0009766.
    The value of y approaches zero as a
    limit, but can never reach zero unless
    the numerator equals zero. But the numerator = 1.

    Let y = 0.
    y = 2^x = 0,
    Take log of both sides:
    xlog2 = log(0),
    X = log(0)/log2,
    But the log of 0 is undefined. Therefore, there is no real value of
    x that will give a y of 0.

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