Question

construa os gráficos das funções exponenciais definidas pelas leis seguintes, destacando seu conjunto imagem:

f(x)=2^x-2​

Answer 1

$\boxed{f(x)=2^x-2}$

$f(x)=0\ \to\ 2^x-2=0\ \therefore\ x=1\ \therefore\ \boxed{(1,0)\in f(x)}$

$x=-1\ \to\ f(-1)=2^{-1}-2=-\dfrac{3}{2}\ \therefore\ \boxed{\bigg(-1,-\dfrac{3}{2}\bigg)\in f(x)}$

$x=0\ \to\ f(0)=2^0-2=1-2=-1\ \therefore\ \boxed{(0,-1)\in f(x)}$

$x=2\ \to\ f(2)=2^2-2=2\ \therefore\ \boxed{(2,2)\in f(x)}$

$x=3\ \to\ f(3)=2^3-2=6\ \therefore\ \boxed{(3,6)\in f(x)}$

$\dots$

$x=6\ \to\ f(6)=2^6-2=62\ \therefore\ \boxed{(6,62)\in f(x)}$

$\dots$

$x=9\ \to\ f(3)=2^9-2=510\ \therefore\ \boxed{(9,510)\in f(x)}$

$x=10\ \to\ f(10)=2^{10}-2=1022\ \therefore\ \boxed{(10,1022)\in f(x)}$

Assíntota horizontal:

$\lim_{x \to -\infty} f(x)= \lim_{x \to -\infty} \big(2^x-2\big)\approx\dfrac{1}{2^{\infty}}-2\approx\boxed{-2}$

$\lim_{x \to \infty} f(x)= \lim_{x \to \infty} \big(2^x-2\big)\approx \boxed{\infty}$

$\boxed{\mathbb{I}\text{m}\big(f(x)\big)=\ ]-1,\infty)}$

O gráfico está em anexo.