The exponential function

The exponential function is the function $f:\mathbb{R}\to (0,\infty ), f(x)={{a}^{x}},\text{ }a>0, a\ne 1$

$f(x)={{a}^{x}},\text{ }a>1$

$\underset{\text{x}\to \infty }{\mathop{\text{lim}}}\,{{a}^{x}}=\infty ; \underset{\text{x}\to -\infty }{\mathop{\text{lim}}}\,{{a}^{x}}=0$

$f(x)={{a}^{x}},\text{ }a\in (0,1)$

$\underset{\text{x}\to \infty }{\mathop{\text{lim}}}\,{{a}^{x}}=0; \underset{\text{x}\to -\infty }{\mathop{\text{lim}}}\,{{a}^{x}}=\infty$

Properties:

1. ${{a}^{x}}>0, (\forall )x\in \mathbb{R}, a>0, a\ne 1$

2. $f(x)={{a}^{x}} \Rightarrow f(0)=1 \Rightarrow \text{punctul }A(0,1)\in Gf(\text{graficului})$

3. $\begin{aligned} & pt\text{ }a>1 f(x)={{a}^{x}}\text{ =strict cresc }\!\!\breve{\mathrm{a}}\!\!\text{ toare} \\ & x<y\Rightarrow {{a}^{x}}<{{a}^{y}} \\ \end{aligned}$

4. $\begin{aligned} & pt\text{ }a\in (0,1) f(x)={{a}^{x}}\text{ =strict descresc }\!\!\breve{\mathrm{a}}\!\!\text{ toare} \\ & x<y\Rightarrow {{a}^{x}}>{{a}^{y}} \\ \end{aligned}$

the exponential function is bijective and invertible

If $a, b\in (0,+\infty )\text{ i }x, y\in \mathbb{R}$, then:

1. ${{a}^{x}}\cdot {{a}^{y}}={{a}^{x+y}}$ 2. $\frac{{{a}^{x}}}{{{a}^{y}}}={{a}^{x-y}}$ 3. ${{\left( \frac{a}{b} \right)}^{x}}={{\frac{a}{{{b}^{x}}}}^{x}}$

4. ${{\left( {{a}^{x}} \right)}^{y}}={{a}^{x\cdot y}}$ 5. ${{\left( a\cdot b \right)}^{x}}={{a}^{x}}\cdot {{b}^{x}}$

Try it

The exponential function, with the base a on a slider.

The chapter states that the function is strictly increasing when a > 1 and strictly decreasing when 0 < a < 1. Drag a through 1 and watch the curve turn over. At exactly a = 1 it flattens into the constant line y = 1, which is why the definition excludes it.

Whatever a is, the curve passes through (0, 1) and never touches the x-axis.