y = e^{-x}

Graph of the Exponential Decay $y = e^{-x}$

$y = e^{-x}$ is a decaying exponential built on $e \approx 2.718$ , and can be written $e^{-x} = \dfrac{1}{e^x} = \left( \dfrac{1}{e} \right)^x$ .

Domain and range

The domain is all real numbers and the range is $y > 0$ ; the value is never zero and never negative. The curve always passes through $(0, 1)$ .

Monotonicity

The derivative is as follows.

y' = -e^{-x}

It is always negative, so the function is monotonically decreasing on the whole line. The second derivative $y'' = e^{-x} > 0$ , so the graph is convex throughout. As $x \to +\infty$ the value approaches $0$ , making the $x$ -axis, the line $y = 0$ , a horizontal asymptote; as $x \to -\infty$ it diverges to $+\infty$ .

Decay by a constant factor

Each time $x$ increases by $1$ , the value is multiplied by $\dfrac{1}{e}$ , about $0.368$ .

$x$	$e^{-x}$	Share of the initial value
$0$	$1$	$100\%$
$1$	$0.368$	about $37\%$
$2$	$0.135$	about $14\%$
$3$	$0.0498$	about $5\%$
$5$	$0.00674$	about $0.67\%$

The value never reaches $0$ , yet it quickly becomes small enough to ignore.

Time constant and half-life

Since one unit step in $x$ multiplies the value by $\dfrac{1}{e}$ , that unit length is called the time constant. The value falls to exactly one half when $x$ increases by $\ln 2 \approx 0.693$ , and that is the half-life. At $(0, 1)$ the tangent has slope $y'(0) = -1$ and is therefore the line $y = 1 - x$ , so near the origin the curve runs downward to the right alongside it.

A geometric sequence made continuous

The values at $x = 0, 1, 2, 3, \ldots$ form a geometric sequence with common ratio $\dfrac{1}{e}$ , so $e^{-x}$ may be read as a geometric sequence joined up smoothly.

Relationships with other functions

It is the mirror image of $y = e^x$ in the $y$ -axis: where $e^x$ grows explosively, $e^{-x}$ decays just as fast. Setting $e^{-x} = t$ gives $x = -\ln t$ , so it is the other face of the logarithm.

Applications

Any process that decreases in proportion to the amount currently present takes the form $e^{-kx}$ .

Radioactive decay
The cooling of a hot body, by Newton's law of cooling
The discharge of a capacitor
The elimination of a drug from the body

It also underlies the exponential distribution in probability.

Graph of the Exponential Decay y=e−xy = e^{-x}y=e−x