Graph of the Function $y = e^x - x$

$y = e^x - x$ is an exponential with a linear function subtracted. The fact that its minimum value is exactly $1$ is precisely the most basic inequality about the exponential function, $e^x \geq x + 1$.

Domain and monotonicity

The domain is all real numbers. The derivative is as follows.

It vanishes when $e^x = 1$, that is at $x = 0$. It is negative for $x < 0$, where $e^x < 1$, and positive for $x > 0$, so the function has a minimum, both local and global, at $x = 0$, of value $e^{0} - 0 = 1$.

Range and the fundamental inequality

Since the minimum is $1$, the range is $y \geq 1$. Rewriting this gives the following, valid for every real $x$.

Equality holds only at $x = 0$. The line $y = x + 1$ is the tangent to $y = e^{x}$ at $x = 0$, so the inequality is an instance of the fact that a concave-up curve lies above its tangent lines. The graph of $y = e^x - x$ can be read as the vertical gap between those two.

Concavity

The second derivative is $y'' = e^{x}$, always positive, so the curve is concave up everywhere and has no inflection point. That convexity is exactly why the tangent-line inequality holds.

Asymptotes

As $x \to -\infty$ the term $e^{x}$ tends to $0$ and $y$ approaches $-x$; indeed $y - (-x) = e^{x} \to 0$, so the line $y = -x$ is a slant asymptote. As $x \to +\infty$, by contrast, $e^{x}$ overwhelms $x$ and the function diverges with no asymptote at all. The result is a markedly asymmetric shape: hugging a straight line on the left, shooting up exponentially on the right.

Counting the solutions of an equation

Because $y \geq 1 > 0$ there is no $x$-intercept, so the equation $e^{x} = x$ has no real solution. More generally the number of solutions of $e^{x} = x + c$ is the number of intersections of this graph with the horizontal line $y = c$.

Consequences

Substituting $\ln t$ for $x$ gives an inequality about the logarithm.

Bounding a logarithm above by a linear function in this way is the starting point of many proofs, including the relation between the arithmetic and geometric means and Gibbs' inequality in information theory. Of all the inequalities about the exponential function, $e^{x} \geq x + 1$ is the one to learn first.

Sample points

The curve passes through $(0, 1)$, $(1, e - 1) \approx (1, 1.718)$, $(-1, e^{-1} + 1) \approx (-1, 1.368)$ and $(2, e^{2} - 2) \approx (2, 5.389)$.