# Is ##f(x)=2^{x}-1## considered an exponential function?

• Callmelucky

#### Callmelucky

Homework Statement
I wonder if it's ##f(x)=2^{x}-1## considered an exponential function?
Relevant Equations
##f(x)=a^{x}##
I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am not sure if the statement "the set of values of an exponential function is a set of positive real numbers" applies only to this type of function ##f(x)=a^{x}## or does it include all types of functions, like this one ##f(x)=2^{x}-1##?

Thank you

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• DeBangis21

Homework Statement:: I wonder if it's ##f(x)=2^{x}-1## considered an exponential function?
Relevant Equations:: ##f(x)=a^{x}##

I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am not sure if the statement "the set of values of an exponential function is a set of positive real numbers" applies only to this type of function ##f(x)=a^{x}## or does it include all types of functions, like this one ##f(x)=2^{x}-1##?

Thank you
Yes, this is an exponential function. Your textbook is considering only functions of the form ##f(x) = a^x##, which would have only positive values. The one you asked about is the translation down by 1 unit of ##y = 2^x##, so the translated version will have negative values when x < 0.

• MatinSAR, DeBangis21 and Callmelucky
Homework Statement:: I wonder if it's ##f(x)=2^{x}-1## considered an exponential function?
Relevant Equations:: ##f(x)=a^{x}##

I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am not sure if the statement "the set of values of an exponential function is a set of positive real numbers" applies only to this type of function ##f(x)=a^{x}## or does it include all types of functions, like this one ##f(x)=2^{x}-1##?

Thank you
This depends more on the context than on a precise definition.

as an algebraic object:
a linear combination of an exponential (##x\mapsto 2^x##) and a constant (##x\mapsto 1##) function

as an algorithmic runtime:
an exponential function, the shift by ##-1## is irrelevant

as an analytical function:
a shifted (by ##c##) exponential function (##x\mapsto a^x+c##)

It is not purely an exponential function, but the effect of minus one is in almost all cases negligible so people might call it exponential despite of it.

• DaveE, MatinSAR, DeBangis21 and 2 others
Yes, this is an exponential function. Your textbook is considering only functions of the form ##f(x) = a^x##, which would have only positive values. The one you asked about is the translation down by 1 unit of ##y = 2^x##, so the translated version will have negative values when x < 0.
thank you

This depends more on the context than on a precise definition.

as an algebraic object:
a linear combination of an exponential (##x\mapsto 2^x##) and a constant (##x\mapsto 1##) function

as an algorithmic runtime:
an exponential function, the shift by ##-1## is irrelevant

as an analytical function:
a shifted (by ##c##) exponential function (##x\mapsto a^x+c##)

It is not purely an exponential function, but the effect of minus one is in almost all cases negligible so people might call it exponential despite of it.
thank you

Ultimately, for ##a>0 ##, you may write : ##a^x =e^{x ln(a)}, ## so I'd say it qualifies.

When considering the growth of the function for large x, I would call it exponential growth. But IMO, to call it an exponential function is a mistake. There are too many situations where you would have to continually mention the added constant.

When considering the growth of the function for large x, I would call it exponential growth. But IMO, to call it an exponential function is a mistake. There are too many situations where you would have to continually mention the added constant.
Still, as ##a ## grows, the value of the function and it's translate will become very close, even if the ln slows the growth of the ## a##

Similar question: is ##f(x) = kx + m ##, ##(m \neq 0 )## considered to be a linear function? Some say "yes" (usually in calculus) because the graph is a straight line.
Some say "no" (usually in linear algebra) because it does not fulfill ##f(x_1 + x_2) = f(x_1) + f(x_2)## and ##f(ax) = af(x)##.

• dextercioby, FactChecker, fresh_42 and 1 other person
Yes, this is an exponential function.
Maybe not an exponential function per se, but definitely a simple transformation of one.

Maybe not an exponential function per se, but definitely a simple transformation of one.
Agreed. The only reason I can think of to not call it officially an exponential function is this. If there are theorems about exponential functions, they might not apply to this function.

Maybe not an exponential function per se, but definitely a simple transformation of one.
It depends on what we consider the essential information and it therewith depends on context. I am used to complexity considerations so ##f(x)=O(2^x).## Others may consider them as linear independent functions in some algebra, ##2^x## and ##-1.## Again others may see its asymptotic behavior, i.e. the exponential part.

The question becomes more interesting if we consider examples like ##f(x)=2^x+x^2+x \log x +c.## Would we still call it exponential? Probably not, although it is still ##f(x)=O(2^x).## So that would be a non-exponential function with an exponential behavior.

• dextercioby and DaveE
Similar question: is ##f(x) = kx + m ##, ##(m \neq 0 )## considered to be a linear function? • 