Why is the exponential function injective but not surjective?

Taniya Melton
2022-10-15
Answered

Why is the exponential function injective but not surjective?

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Layne Murillo

Answered 2022-10-16
Author has **14** answers

The real valued function $f=\mathrm{exp}:\mathbb{R}\to \mathbb{R}$ has the properties that f(0)=1, f′=f and f(x+y)=f(x)f(y) for all $x,y\in \mathbb{R}$. Thus, 1=f(0)=f(x+(−x))=f(x)f(−x) and in particular, $f(x)\ne 0$ for all $x\in \mathbb{R}$. So, f is not surjective. Since f is continuous, it thus also follows from the intermediate value theorem that f either attains only positive values or only negative values. As f(0)=1, it follows that f(x)>0 for all $x\in \mathbb{R}$. Now, it follows that f′(x)=f(x)>0, and thus f is strictly increasing in $\mathbb{R}$. Every strictly increasing function is injective, thus f is injective.

Interestingly, the exponential function can be extended to $\mathrm{exp}:\mathbb{C}\to \mathbb{C}$, where the function is no longer injective, and attains all complex values except for the sole exception of 0.

Interestingly, the exponential function can be extended to $\mathrm{exp}:\mathbb{C}\to \mathbb{C}$, where the function is no longer injective, and attains all complex values except for the sole exception of 0.

asked 2022-06-24

asked 2022-11-06

Exponential functions always have a linear variable as a power of constant base and an increasing monotonic graph. Are these ${e}^{\surd x}$ and ${e}^{x}$ functions also exponential functions. What is the unique characteristic for which a function is said exponential and do these functions has the same traits. Why they do not have a monotonic graph? Which ${e}^{x}$ has. Do they need to satisfy the pre defining properties an exponential function should satisfy i.e. ( base>0, base must not be 1, and power always belongs to real set (R)). Please clarify over each point.

asked 2022-07-26

Write exponential functions in the form of $y=a{e}^{kt}$

find k accurate to four decimal places. If t is measured inyears, give the percent annual growth rate and continuous growthrate per year.

a) a city is growing by 26% per year.

b)a company's profit is increasing by an annual growth factor of 1.12

find k accurate to four decimal places. If t is measured inyears, give the percent annual growth rate and continuous growthrate per year.

a) a city is growing by 26% per year.

b)a company's profit is increasing by an annual growth factor of 1.12

asked 2022-11-24

the equation f(x)′−f(x)=0 holds for the exponential function on the complex plane.Now what i dont understand is this.

"let $f(x)={a}_{0}+{a}_{1}X+{a}_{2}{X}^{2}........$ f is a polynomial with infinite degree ".Why is that. I dont understand how he came to that conclusion?I mean Why define it that way?.MAybe he could solve the ODE on the real numbers and avoid this "out of nowhere" polynomial or is there a connection?

"let $f(x)={a}_{0}+{a}_{1}X+{a}_{2}{X}^{2}........$ f is a polynomial with infinite degree ".Why is that. I dont understand how he came to that conclusion?I mean Why define it that way?.MAybe he could solve the ODE on the real numbers and avoid this "out of nowhere" polynomial or is there a connection?

asked 2022-10-08

Solve the exponential equation ${32}^{x+3}={4}^{3x+5}$ for x

{-4}

{5}

{}

{-1}

{-4}

{5}

{}

{-1}

asked 2022-11-15

Is there a way to transform the function

exp(A+B+C),

where $\mathrm{exp}(\cdot )$ is the exponential function, into a sum

f(A)+f(B)+f(C)?

exp(A+B+C),

where $\mathrm{exp}(\cdot )$ is the exponential function, into a sum

f(A)+f(B)+f(C)?

asked 2022-10-20

What is the difference between this notation of the exponential function

$$(1+\frac{1}{n}{)}^{n}\to e\phantom{\rule{0ex}{0ex}}\mathbf{a}\mathbf{s}\phantom{\rule{0ex}{0ex}}n\to \mathrm{\infty}$$

and this notation:

$$\underset{n\to \mathrm{\infty}}{lim}(1+\frac{x}{n}{)}^{n}$$

Why is there a variable x in the second equation, and a 1 in the first equation?

$$(1+\frac{1}{n}{)}^{n}\to e\phantom{\rule{0ex}{0ex}}\mathbf{a}\mathbf{s}\phantom{\rule{0ex}{0ex}}n\to \mathrm{\infty}$$

and this notation:

$$\underset{n\to \mathrm{\infty}}{lim}(1+\frac{x}{n}{)}^{n}$$

Why is there a variable x in the second equation, and a 1 in the first equation?