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I saw two different forms for an exponential function which are:

  • $f(x)=a^x$

and

  • $f(x)=a \cdot b^x$ where $a$ is the initial value

Are the rules and cases the same in both forms such as:

  • It is always greater than 0, and never crosses the x-axis
  • It always intersects the y-axis at y=1 ... in other words it passes through (0,1)
  • At x=1, f(x)=a ... in other words it passes through (1,a)
  • It is an Injective (one-to-one) function
  • Its Range is the Positive Real Numbers: (0, +∞)

and so on.

Steve
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  • "... where $a$ is the initial value" looks like a big hint for whether or not the second condition holds in both cases. –  Sep 01 '17 at 11:46
  • They are not the same, the second one is a general form of the first one (which is the particular case where the multiplying factor is $1$). Also, the second and third "rules" are not true for the second form: the intersection is at $y=a$ and $f(1)=a,b$ – Daniel Cunha Sep 01 '17 at 11:48
  • Additional note: You can always write $a \times b^x = b^{\log_b{a}} b^x = b^{x+\log_b{a}}$. – Matti P. Sep 01 '17 at 11:49
  • @G.Sassatelli Also the third condition suffers a change of scale. – Miguel Sep 01 '17 at 11:50

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