Question

Which of the following are TRUE regarding ​f(x) = a(b)^x?

a) The exponential function has a y-intercept at (0, 1).
b) As x increases, the function approaches positive infinity if a and b are positive.
c) As x approaches negative infinity, the function approaches zero if a and b are positive.
d) The exponential function is always decreasing.

Answer 1

The properties of the exponential function f(x) = a(b)^x include it approaching positive infinity as x increases, and approaching zero as x approaches negative infinity, both when a and b are positive. The function is not always decreasing; it depends on the value of b.

Step-by-step explanation:

The question concerns the properties of the exponential function f(x) = a(b)^x. Let's analyze each statement:

(a) The exponential function has a y-intercept at (0, a), not necessarily (0, 1) unless a = 1.

(b) As x increases, the function approaches positive infinity if a and b are positive.

(c) As x approaches negative infinity, the function approaches zero if a and b are positive.

(d) The exponential function is always decreasing only if 0 < b < 1; otherwise, it is increasing if b > 1.

Statement (b) and (c) are true, while (a) is only true if a = 1, and (d) is not necessarily true as it depends on the value of b.