Question

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

Answer 1

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.