In the exponential function f(x) = a b^x, which conditions must hold for a and b?

In the exponential function f(x) = a b^x, which conditions must hold for a and b?

• A. a must be zero and b positive
• B. a ≠ 0, b > 0, and b ≠ 1
• C. a ≥ 0 and b ≥ 0
• D. a > 1 and b < 0

Answer: (B)

The key idea is that in an exponential function f(x) = a b^x, a sets the starting value at x = 0 and b shapes the growth or decay, so to get a true real exponential function you need three things: a is not zero, the base b is positive, and the base is not equal to 1. If a were zero, the function would be identically zero for all x, which is not an exponential curve. If the base were not positive, b^x isn’t well-defined as a real number for all real x (a negative base would give complex values for most x, and 0 as a base is problematic). If the base were 1, the function would be constant, not exponential growth or decay. With a nonzero a, a positive base b different from 1, you get a genuine exponential function that either grows (b > 1) or decays (0 < b < 1) and passes through the point (0, a).

The key idea is that in an exponential function f(x) = a b^x, a sets the starting value at x = 0 and b shapes the growth or decay, so to get a true real exponential function you need three things: a is not zero, the base b is positive, and the base is not equal to 1.

If a were zero, the function would be identically zero for all x, which is not an exponential curve. If the base were not positive, b^x isn’t well-defined as a real number for all real x (a negative base would give complex values for most x, and 0 as a base is problematic). If the base were 1, the function would be constant, not exponential growth or decay. With a nonzero a, a positive base b different from 1, you get a genuine exponential function that either grows (b > 1) or decays (0 < b < 1) and passes through the point (0, a).