Question

). find the domain and range of the
your answers look good! can you explain why the domain
includes all real numbers? what property of exponential
functions supports this?
ask the al coach a question

Explanation:

For an exponential function of the form \( f(x) = a^x \) (where \( a>0, a
eq1 \)), the domain includes all real numbers because we can raise \( a \) to any real - valued exponent. For example, if \( x \) is a positive integer, we know how to calculate \( a^x \) (repeated multiplication). If \( x \) is a negative integer, \( a^x=\frac{1}{a^{\vert x\vert}} \). If \( x \) is a fraction (rational number), say \( x = \frac{m}{n} \) (in lowest terms), \( a^x=\sqrt[n]{a^m} \) (for real - valued roots, we need \( a\geq0 \) when \( n \) is even, but in the general definition of exponential functions with \( a > 0,a
eq1 \), this is well - defined). For irrational numbers, we can use the concept of limits of sequences of rational numbers approaching the irrational number to define \( a^x \). In short, there are no restrictions on the real - valued input \( x \) for an exponential function in terms of what real number \( x \) can be, so the domain is all real numbers. The property that supports this is the well - defined nature of exponentiation for any real number when the base \( a>0,a
eq1 \).

Answer:

The domain of an exponential function \( f(x)=a^x\) (\(a > 0,a
eq1\)) includes all real numbers because exponentiation with base \(a>0,a
eq1\) is well - defined for any real - valued exponent. The property of exponential functions that supports this is the ability to raise the base \(a\) (where \(a>0,a
eq1\)) to any real number \(x\) (positive, negative, rational, irrational) and get a well - defined real (or complex, but for real - valued exponential functions with \(a>0\)) output.