Question

consider the function y = 2 .
5c how would you describe the rate of increase of the function?
the correct option was c
as x increases, the function increases at a slower and slower rate.
as x increases, the function increases at a constant rate.
as x increases, the function increases at a faster and faster rate. c
5 nailed it!
5d what is the domain of the function?
x > 0
a all real x
b x ≥ 0
c x < 0
d
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Explanation:

For the function \(y = 2^x\), the rate of increase is determined by its derivative. The derivative of \(y = 2^x\) is \(y'=2^x\ln 2\). As \(x\) increases, \(2^x\) increases exponentially. Since \(\ln 2>0\), \(y'\) (the rate of change) also increases exponentially. So, as \(x\) increases, the function \(y = 2^x\) increases at a faster and faster rate.

For the domain of the function \(y = 2^x\), the exponential function \(a^x\) (where \(a = 2>0\)) is defined for all real values of \(x\). There is no restriction on the value of \(x\) for which the function \(y = 2^x\) is not well - defined.

Answer:

For the rate of increase: As \(x\) increases, the function increases at a faster and faster rate.
For the domain: all real \(x\)