What is an exponential function?

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What is an exponential function?

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Answer 1 · 2015-03-23T00:17:42

The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable.

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The function is often written as exp(x) It is widely used in physics, chemistry, engineering, mathematical biology, economics and mathematics.

Answer 2 · 2015-03-23T00:32:34

An exponential function is a function of the form $f (x) = a^{x}$ $f(x)=a^x$ with $a > 0$ $a>0$ but $a \neq 1$ $a!=1$ .

For integer and rational $x$ $x$ , we give definitions of $a^{x}$ $a^x$ earlier in algebra classes.
For irrational $x$ $x$ we owe you a definition, but one approach to the definition is to describe $a^{x}$ $a^x$ for irrational $x$ $x$ as the number thar $a^{r}$ $a^r$ gets closer to as rational $r$ $r$ get close to $x$ $x$ . (We owe you a proof that there is a unique such number.)

Examples:

$f (x) = 2^{x}$ $f(x)=2^x$

$f (x) = 5^{x}$ $f(x)=5^x$

$f (x) = {(\frac{2}{5})}^{x}$ $f(x)=(2/5)^x$

$f (x) = 4^{x} = {(2^{2})}^{x} = 2^{2 x}$ $f(x)=4^x=(2^2)^x = 2^(2x)$

The last example illustrates why we also consider $f (x) = a^{k x}$ $f(x)=a^(kx)$ for constant $k \neq 0$ $k!=0$ to be exponential functions

We can write $f (x) = a^{k x} = {(a^{k})}^{x}$ $f(x)=a^(kx)=(a^k)^x$ and .for $a > 0$ $a>0$ and $k \neq 0$ $k!=0$ this $f (x) = b^{x}$ $f(x)=b^x$ for the "right kind" of $b$ $b$ . ( $b > 0$ $b>0$ , and $b \neq 1$ $b!=1$ )

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What is an exponential function?

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