Exponential function y = aˣ. Graphs and properties (for a > 1 and 0 < a < 1)

Exponential function

An exponential function has the form $y=a^x$ , where $a$ is a fixed positive number, $a\neq1$ , and $x$ is the independent variable (the exponent). In school, a positive base $a>0$ matches the meaning of $a^x$ for any real $x$ : for $x=n\in\mathbb{Z}$ you recover integer powers; for fractional $x$ — roots and powers from earlier sections. Domain: $D(y)=\mathbb{R}$ . Range: $E(y)=(0;+\infty)$ — an exponential never hits zero and never takes negative values. Mandatory point: $(0;1)$ , since $a^0=1$ for $a>0$ .

Base

a=1

is excluded:

1^x=1

is flat, not exponential in the monotonicity sense

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