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

[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

 

[Mark44]

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.

 

[fresh_42]

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.

 

[Callmelucky]

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

 

[Callmelucky]

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

 

[WWGD]

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

 

[FactChecker]

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.

 

[WWGD]

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##

 

[malawi_glenn]

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

It depends on your definition.
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)##.

 

[Mark44]

Yes, this is an exponential function.

Maybe not an exponential function per se, but definitely a simple transformation of one.

 

[FactChecker]

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.

 

[fresh_42]

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.

 

[WWGD]

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

It depends on your definition.
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)##.

That's a(ffine) example you used.