Question

Which statements are true about exponential functions? Check all that apply.

The domain is all real numbers.
The range always includes negative numbers.
The graph has a horizontal asymptote at x = 0.
The input to an exponential function is the exponent.
The base represents the multiplicative rate of change.

Answer 1

True: The domain is all real numbers.
- Exponential functions of the form b^x (b>0, b≠1) accept any real x, and vertical shifts or horizontal shifts do not introduce domain restrictions.

False: The range always includes negative numbers.
- Not always. Basic exponentials b^x are strictly positive (range (0, ∞)). A negative coefficient can make the range negative, but it is not always the case.

False: The graph has a horizontal asymptote at x = 0.
- That statement is wrong as written: x = 0 is a vertical line. The usual horizontal asymptote for b^x is y = 0 (or y = c if the function is vertically shifted).

True: The input to an exponential function is the exponent.
- The independent variable appears in the exponent (e.g., b^x), which is the defining feature.

True: The base represents the multiplicative rate of change.
- Increasing x by 1 multiplies the value by the base b (b^{x+1} = b·b^x), so b is the per-unit multiplicative factor.