Exercícios sobre definição de função

Seja $\;f\,:\, \mathbb{R} \rightarrow \mathbb{R} \;$ a função que a cada número real não nulo associa a soma dele com o seu inverso. Calcule:

$\,f(2)\,$

$\,f(\dfrac{1}{2})\,$

$\,f(x)\,$

$\,f(\dfrac{1}{x})\,$

$\,f(x\,+\,1)\,$

$\,f(x\,-\,1)\,$

resposta: a) f(2) = 2 + 1/2 = 5/2

b) $f(\dfrac{1}{2})\,=\,\dfrac{1}{2}\,+\,$$\dfrac{1}{\frac{1}{2}}\,=\,\dfrac{1}{2}\,+\,2\,=\,$ $\dfrac{5}{2}$

c) $f(x)\,=\,x\,+\,\dfrac{1}{x}\,=\,$ $\dfrac{x^2\,+\,1}{x}$

d) $f(\dfrac{1}{x})\,=\,\dfrac{1}{x}\,+\,\dfrac{1}{\frac{1}{x}}\,$ $=\,\dfrac{1}{x}\,+\,x\,=$ $\,\dfrac{x^2\,+\,1}{x}$

e)$f(x\,+\,1)\,=$ $\,(x\,+\,1)\,+\,\dfrac{1}{x\,+\,1}\,=\,\dfrac{(x\,+\,1)^2\,+\,1}{x\,+\,1}\,=\,$$\dfrac{x^2\,+\,2x\,+\,2}{x\,+\,1}$

f)$f(x\,+\,1)\,=$ $\,(x\,-\,1)\,+\,\dfrac{1}{x\,-\,1}\,=\,\dfrac{(x\,-\,1)^2\,+\,1}{x\,-\,1}\,=\,$$\dfrac{x^2\,-\,2x\,+\,2}{x\,-\,1}$