Is g(x)=5^sqrt(x) an Exponential Function?

[drewfstr314]

Our textbook defines an exponential function as
f(x) = ab^x. However, a question was brought up about a function, g(x) = 5^sqrt(x). Is g an exponential function? It looks like an exponential graph for x>0, but is not continuous on R.

Thanks in advance!

 

[dextercioby]

A function is not defined in the absence of domain and range. So do tell us the whole definition as worded by your book.

 

[jbriggs444]

Our textbook defines an exponential function as
f(x) = ab^x. However, a question was brought up about a function, g(x) = 5^sqrt(x). Is g an exponential function? It looks like an exponential graph for x>0, but is not continuous on R.

Let us ignore the fact that g(x) = 5^sqrt(x) is not always real-valued for negative x and ask the somewhat better defined question:

"Does the graph of g(x) = 5^sqrt(x) match the graph of any function f(x) of the form ab^x for real-valued constants a and b and positive real x"

One simple-minded way to answer this would be to assume that there is such a function and realize that, if so:

f(0) = a*b^0 = a = g(0) = 5^sqrt(0) = 1

So a = 1

f(1) = a*b^1 = ab = b = g(1) = 5^sqrt(1) = 5

So b = 5

So the question then becomes:

"does the graph of g(x) = 5^sqrt(x) match the graph of f(x) = 5^x"

The answer to that question is rather obvious.