Question

\(y=a^x\), where a is greater than zero but not equal to 1 and x is an exponent

Answer 1

This is an exponential function where the base is a and the exponent is x. As x increases, the value of y increases exponentially. When a is greater than 1, the function will have a vertical asymptote at x=0 and will approach y=0 as x approaches negative infinity. When a is between 0 and 1, the function will approach y=0 as x approaches positive infinity and will have a horizontal asymptote at y=0. If a is less than 0, the function will be undefined for all real values of x.

Answer 2

If a is less than 0, the function will be undefined for all real values of x.
This is not always true.
(-4)^2 = 16
(-2.378)^-3 = -0.07436...

The statement is "true" only if you define a^x = e^(x ln(a))

Answer 3

You are correct. When a is a negative number, the expression a^x is not a real number for all values of x, unless x is an integer and a is an odd number. However, if we extend the concept of exponentiation to include complex numbers, then we can define a^x for all x in the complex plane and a in the set of real numbers, including negative numbers. In this case, we define a^x as e^(xln(a)), where ln(a) denotes the natural logarithm of a.