Properties of exponential function and its graph

Exponential function $y=a^x$ , $a>0$ , $a\neq 1$

In Grade 11 you treat $a^x$ as a structured object for equations, inequalities, and problems with parameters: • Domain: $D=\mathbb{R}$ (the exponent $x$ is any real number). • Range: $E=(0,+\infty)$ — for $a>0$ , $a\neq1$ the power never hits 0 and is never negative on $\mathbb{R}$ . • Monotonicity: if $a>1$ the function strictly increases; if $0<a<1$ it strictly decreases. This is the backbone for inequalities: with $a>1$ the inequality sign is preserved in $x$ ; with $0<a<1$ it flips. • Horizontal asymptote: the line $y=0$ as $x\to-\infty$ (if $a>1$ ) and as $x\to+\infty$ (if $0<a<1$ ).

Before using $a^x$ in an equation, check $a>0$ , $a\neq1$

At a glance

Base $a$	Behaviour of $a^x$
$a>1$	growth; $x_1<x_2\Rightarrow a^{x_1}<a^{x_2}$
$0<a<1$	decay; $x_1<x_2\Rightarrow a^{x_1}>a^{x_2}$
$a=1$	in school $1^x=1$ is not treated as a separate “constant-base exponential” theory

💡Next lesson — exponential equations: same base, substitution $t=a^x$ , taking logarithms.

✅You rely on $D$ , $E$ , monotonicity, and the asymptote $y=0$ for $y=a^x$ — the foundation of the whole section.