Properties of the exponential function

Let us collect in a single list all the properties of the exponential function: they are the same for every admissible base, with the only difference between growth ($a>1$) and decay ($0<a<1$).

Properties — The exponential function

For every admissible base $a>0,\ a\ne 1$:

• Domain: $\mathbb{R}$.
• Image: $(0;+\infty)$, that is the function is always strictly positive.
• Intersections: $\exp_a(0) = a^0 = 1$, so the graph passes through $(0;1)$. There is no intersection with the $x$-axis (a recurring mistake among students).
• Monotonicity: strictly increasing if $a > 1$, strictly decreasing if $0 < a < 1$.
• Injectivity: strict monotonicity implies injectivity; since the image $(0;+\infty)$ coincides with the codomain, the function is a bijection from $\mathbb{R}$ onto $(0;+\infty)$ and therefore admits an inverse (the logarithm, of the next chapter).
• Asymptotes: the $x$-axis ($y=0$) is a horizontal asymptote. If $a>1$ it is so towards $-\infty$; if $0<a<1$ it is so towards $+\infty$.

The property most used in exercises is injectivity: from $a^{f(x)}=a^{g(x)}$ one can deduce $f(x)=g(x)$, that is “remove the base”. It is precisely the bijectivity that also guarantees the existence of the inverse function, the logarithm.

Links

Topics: Exponential function
Concepts: Asymptote · Bijectivity · Domain · Inverse function · Image · Injectivity · Monotonicity
Functions: Exponential function
Skills: Analysing a function