essential question 2

Q.1
For a＞1,
y = f(x) = a^x is strictly increasing,
with ＋ve slope increasing with x.
　When x→-∞, y→0 (y is always ＋ve).
　When x = 0, y = 1.
　When x→+∞, y→+∞.
y = g(x) = log{base a}x is strictly increasing,
with ＋ve slope decreasing (but always ＋ve slope) with x.
　When x→0 (x is always ＋ve), y→-∞.
　When x = 1, y = 0.
　When x→+∞, y→+∞.

For 0＜a＜1,
y = f(x) = a^x is a mirror image of "y = f(x)" above, mirrored about y-axis.
y = g(x) = log{base a}x is a mirror image of "y = g(x)" above, mirrored about x-axis.

2007-02-03 00:01:10 補充：
Q.2y = f(x) = a^x　is equivalent to　x = log{base a} y = g(y).y = g(x) = log{base a} x　is equivalent to　x = a^y = f(y).Therefore, y = f(x) can be inter-changed with y = g(x), by inter-changing the x- and y-axes.

2007-02-03 00:01:24 補充：
In other words, the graph of y = g(x) is symmetric to that of y = f(x), mirrored about the line y = x.Therefore, the intersection points of y=f(x) and y=g(x) must be on the line y = x.

2007-02-03 00:31:15 補充：
That means,- if y=f(x) cuts y=x at 2 points, that means y=f(x) intersects twice with y=g(x).- if y=f(x) touches y=x at 1 point, that means y=f(x) touches y=g(x) once.- if x＜f(x) for all x, that means y=f(x) does not intersect with y=g(x).