Graph of the Exponential Function $y = 2^x$

$y = 2^x$ is the exponential function with base $2$: its value doubles each time $x$ increases by $1$, making it the archetype of repeated doubling.

Domain and range

The domain is all real numbers. Since $2^x$ is never zero and never negative, the range is $y > 0$. Negative exponents simply give small positive values, such as $2^{-1} = \dfrac{1}{2}$ and $2^{-2} = \dfrac{1}{4}$.

Monotonicity

The derivative is as follows.

Here $\ln 2 \approx 0.693$ is a positive constant and $2^x > 0$, so $y'$ is always positive and the function increases monotonically. The second derivative $y'' = 2^x (\ln 2)^2$ is positive as well, so the curve is convex throughout: the larger it grows, the faster it grows.

Asymptotes and limits

As $x \to -\infty$, $2^x \to 0$, so the $x$-axis, the line $y = 0$, is a horizontal asymptote. The values stay positive, so the curve approaches it from above. As $x \to +\infty$ the function diverges to $+\infty$.

Repeated doubling

Because $2^{x+1} = 2 \cdot 2^x$, each unit step in $x$ multiplies the value by exactly $2$.

Ten steps multiply the value by $2^{10} = 1024$, roughly a thousandfold. Modest at first, it eventually outgrows every polynomial: for large enough $x$, $2^x$ overtakes even $x^{100}$.

Notable points

The curve passes through $(0, 1)$, $(1, 2)$, $(2, 4)$ and $\left( -1, \dfrac{1}{2} \right)$. Since $a^0 = 1$, every exponential function passes through $(0, 1)$ whatever its base.

How the base changes the curve

The inverse is $\log_2 x$. Writing $2^x = e^{x \ln 2}$ shows that it is $y = e^x$ compressed horizontally.

Indeed $\left( \dfrac{1}{2} \right)^x = 2^{-x}$ is this graph reflected in the $y$-axis, a decreasing curve.

Applications

It models anything that doubles.

• A cell divides into two, so after $n$ divisions there are $2^n$ cells
• In binary, $n$ bits represent $2^n$ distinct values
• Halving processes such as radioactive decay are written $2^{-t/T}$

The essence of exponential growth is that the time it takes to double stays the same, no matter how large the quantity already is.