Capítulo 1 Funções e grá cos

❈❛♣ít✉❧♦ ✶

❋✉♥çõ❡s ❡ ❣rá✜❝♦s

❉❡✜♥✐çã♦ ✶✳ ❙❡❥❛♠ X ❡ Y ❞♦✐s s✉❜❝♦♥❥✉♥t♦s ♥ã♦ ✈❛③✐♦s ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❯♠❛ ❢✉♥çã♦ ❞❡ X ❡♠ Y ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✉♠❛ ❢✉♥çã♦ é ✉♠❛ r❡❣r❛✱ ❧❡✐ ♦✉ ❝♦♥✈❡♥çã♦ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ x ❞♦ ❝♦♥❥✉♥t♦ X ✉♠ ú♥✐❝♦ ❡❧❡♠❡♥t♦ y ❞♦ ❝♦♥❥✉♥t♦ Y. ❖ ❝♦♥❥✉♥t♦ X é ❝❤❛♠❛❞♦ ❞❡ ❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦ ❡ ♦ ❝♦♥❥✉♥t♦ Y ❞❡ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ❢✉♥çã♦✳

❯s✉❛❧♠❡♥t❡ ❞❡♥♦t❛♠♦s ❛ r❡❣r❛ ♣❡❧♦ sí♠❜♦❧♦x_7−→y❡ ✐♥❞✐❝❛♠♦s ❛ ❢✉♥çã♦ ♣♦r ✉♠❛ ❧❡tr❛ ❧❛t✐♥❛ ♠✐♥ús❝✉❧❛✳ ❆ss✐♠✱ ❞❡♥♦t❛♠♦s ✉♠❛ ❢✉♥çã♦ ♣❡❧♦ sí♠❜♦❧♦

f :X _−→Y, x_7−→y.

◆❡st❡ ❝❛s♦✱ ✐♥❞✐❝❛✲s❡ ♦ ❞♦♠í♥✐♦ ❞❡ f ♣❡❧♦ sí♠❜♦❧♦ Df(= X) ❡ ♦ ú♥✐❝♦

❡❧❡♠❡♥t♦ y, ❞❡ Y, ❛ss♦❝✐❛❞♦ ❛♦ ❡❧❡♠❡♥t♦ x, ❞❡ X, ♣❡❧♦ sí♠❜♦❧♦ f(x) ✭❧ê✲

s❡✿ ✧f ❞❡ x✧✮✱ ❝❤❛♠❛❞♦ ❞❡ ❡❧❡♠❡♥t♦ ✐♠❛❣❡♠ ❞❡ f ❡♠ x ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❛ ✐♠❛❣❡♠ ❞❡ f ❡♠ x. ❉✐r❡♠♦s q✉❡ x é ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ f ❡ q✉❡y é ❛ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ ❞❡ f.

◆♦t❛çã♦ ✶✳ ❯♠❛ ♦✉tr❛ ♥♦t❛çã♦ ♣❛r❛ ✉♠❛ ❢✉♥çã♦ ❢ é ❞❛❞❛ ♣♦r✿ f :Df 7−→Y, x 7−→y =f(x).

❉❡✜♥✐çã♦ ✷✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ f : X(= Df) 7−→ Y, x 7−→ y = f(x).

❉❡✜♥✐♠♦s ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ f ❝♦♠♦ ♦ s✉❜❝♦♥❥✉♥t♦✱ ❞♦ ❝♦♥❥✉♥t♦ ♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ X_×Y, ❞❛❞♦ ♣♦r

Gf ={(x, y) | y =f(x), ♣❛r❛x∈X}.

❉❡✜♥✐çã♦ ✸✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ f : X(= Df) 7−→ Y, x 7−→ y = f(x).

❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❞❡ f ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❛ ✐♠❛❣❡♠ ❞❡ f ❝♦♠♦ ♦ s✉❜❝♦♥❥✉♥t♦✱ ❞♦ ❝♦♥tr❛❞♦♠í♥✐♦ Y, ❞❛❞♦ ♣♦r

Im(f) =_{y_∈Y _| y=f(x), ♣❛r❛ x_∈X_}.

✷ ❈❆P❮❚❯▲❖ ✶✳ ❋❯◆➬Õ❊❙ ❊ ●❘➪❋■❈❖❙

❋✐❣✉r❛ ✶✳✶✿ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❝♦♠ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê❡♥❝✐❛ ②❂❢✭①✮✳

❆ss✐♠✱ y_∈Im(f) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ x_∈Df t❛❧ q✉❡ y =f(x).

❉❡✜♥✐çã♦ ✹✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ f : X(= Df) 7−→ Y, x 7−→ y = f(x).

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛ ❡♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ I ❞❡ Df

♦✉ s✐♠♣❧❡s♠❡♥t❡ q✉❡ f é ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛ ❡♠ ■✱ s❡

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1, x2 ∈I t❛✐s q✉❡ x1 6=x2, ❡♥tã♦f(x1)6=f(x2),

♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ s❡

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1, x2 ∈I t❛✐s q✉❡ f(x1) = f(x2), ❡♥tã♦ x1 =x2.

◗✉❛♥❞♦ I =Df, ❞✐r❡♠♦s q✉❡ f é ✐♥❥❡t♦r❛✳

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ s♦❜r❡❥❡t♦r❛ s❡ ♦ ❝♦♥❥✉♥t♦ ✐♠❛❣❡♠ ❞❡ f ❝♦✐♥❝✐❞❡ ❝♦♠ ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❡ f, ✐st♦ é✱ s❡

Im(f) =Y, ♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ s❡

♣❛r❛ t♦❞♦ y_∈Y, ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ x_∈Df t❛❧ q✉❡ y=f(x).

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛ s❡ ❡❧❛ é ✉♠❛ ❢✉♥ççã♦ ✐♥❥❡t♦r❛ ❡ s♦❜r❡❥❡t♦r❛✳

Pr♦♣♦s✐çã♦ ✶✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ f :X(=Df)7−→Y, x 7−→y=f(x). ❙❡ f

é ❜✐❥❡t♦r❛✱ ❡♥tã♦ ❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❛❞❛ ♣♦r g :Y =Im(f)

−→X(= Df), y 7−→x=g(y),

♦♥❞❡ x = g(y) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ y = f(x), ❞❡✜♥❡ ✉♠❛ ❢✉♥çã♦ ❞❡ Y ❡♠ X, ❞❡ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ y ❡ ✈❛r✐á✈❡❧ ❞❡♣❡♥❞❡♥t❡ x.

✸

f−1 :Y −→X, y 7−→x=f−1(y),

♦♥❞❡ x=f−1(y)s❡✱ ❡ s♦♠❡♥t❡ s❡✱ y=f(x).

❋✐❣✉r❛ ✶✳✷✿ ❘❡❧❛çã♦ ❞♦s ♣♦♥t♦s ❞♦ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❢ ❝♦♠ ❛ s✉❛ ✐♥✈❡rs❛ f−1✳

◆♦t❛çã♦ ✷✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛f :X(=Df)−→Y =Im(f)

, x_7−→ y = f(x). ❯s✉❛❧♠❡♥t❡✱ ❡s❝r❡✈❡♠♦s ❛ ❢✉♥çã♦ ✐♥✈❡rs❛✱ ❞❡ f, t♦♠❛♥❞♦ ❛ ❧❡tr❛ x ❝♦♠♦ s✉❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✱ ❡♠ ❧✉❣❛r ❞❡ y, ❡ y ❝♦♠♦ s✉❛ ✈❛r✐á✈❡❧ ❞❡✲ ♣❡♥❞❡♥t❡✱ ❡♠ ❧✉❣❛r ❞❡ x. ❆ss✐♠✱ ❡s❝r❡✈❡r❡♠♦s ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❡ f ❝♦♠♦

f−1 :Y −→X, x7−→y=f−1(x),

♦♥❞❡ y=f−1(x) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x=f(y).

❉❡✜♥✐çã♦ ✺✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦f :X(=Df)−→Y, x−→y =f(x)❡I ⊆Df

✉♠ ✐♥t❡r✈❛❧♦✳ ❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡♠ I s❡ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1, x2 ∈I t❛✐s q✉❡ x1 < x2, ❡♥tã♦ f(x1)< f(x2).

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ ❞❡❝r❡s❝❡♥t❡ ❡♠ I s❡

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1, x2 ∈I t❛✐s q✉❡ x1 < x2, ❡♥tã♦ f(x1)> f(x2).

◗✉❛♥❞♦ I =Df, ❞✐r❡♠♦s q✉❡ f é ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ ♥ã♦✲❝r❡s❝❡♥t❡ ❡♠ I s❡

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1, x2 ∈I t❛✐s q✉❡ x1 < x2, ❡♥tã♦ f(x1)≥f(x2).

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❢ é ✉♠❛ ❢✉♥çã♦ ♥ã♦✲❞❡❝r❡s❝❡♥t❡ ❡♠ I s❡

q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ x1, x2 ∈I t❛✐s q✉❡ x1 < x2, ❡♥tã♦ f(x1)≤f(x2).

◗✉❛♥❞♦ I =Df, ❞✐r❡♠♦s q✉❡ f é ♥ã♦✲❝r❡s❝❡♥t❡ ♦✉ ♥ã♦✲❞❡❝r❡s❝❡♥t❡✱ r❡s♣❡❝✲

✹ ❈❆P❮❚❯▲❖ ✶✳ ❋❯◆➬Õ❊❙ ❊ ●❘➪❋■❈❖❙

❋✐❣✉r❛ ✶✳✸✿ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✳

Pr♦♣♦s✐çã♦ ✷✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ f : X(= Df) −→ Y, x −→ y = f(x) ❡

I _⊆Df ✉♠ ✐♥t❡r✈❛❧♦✳ ❙❡ f é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡ ❡♠ I ♦✉ ❞❡❝r❡s❝❡♥t❡ ❡♠

I, ❡♥tã♦ f é ✉♠❛ ❢✉♥çã♦ ✐♥❥❡t♦r❛ ❡♠ I.

❋✐❣✉r❛ ✶✳✹✿ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❞❡❝r❡s❝❡♥t❡✳

❉❡✜♥✐çã♦ ✻✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ f : X(= Df) −→ Y, x 7−→ y = f(x), t❛❧

q✉❡ ♦ ❞♦♠í♥✐♦ Df ✈❡r✐✜q✉❡ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿ ♣❛r❛ t♦❞♦ x ∈ Df, t❡♠♦s

−x_∈Df.

✺

❉✐③❡♠♦s q✉❡ ❛ ❢✉♥çã♦ f é ✉♠❛ ❢✉♥çã♦ ✐♠♣❛r s❡✱ f(₋x) = ₋f(x), ♣❛r❛ t♦❞♦ x_∈Df.

❋✐❣✉r❛ ✶✳✺✿ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣❛r✳

❋✐❣✉r❛ ✶✳✻✿ ●rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ✐♠♣❛r✳

❉❡✜♥✐çã♦ ❞❡ ❢✉♥çã♦ ❝♦♠♣♦st❛

❉❡✜♥✐çã♦ ✼✳ ❙❡❥❛♠ ❢✉♥çõ❡s

✻ ❈❆P❮❚❯▲❖ ✶✳ ❋❯◆➬Õ❊❙ ❊ ●❘➪❋■❈❖❙

❡

g :Dg ⊆Y →Z, y7→z =g(y),

✈❡r✐✜❝❛♥❞♦ ❛ ❝♦♥❞✐çã♦Im(f)_⊆Dg.

❉❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❝♦♠♣♦st❛✱ ❞❡ f ❝♦♠ g, ♣♦r

g_◦f :Df ⊆X →Z, x7→z = (g◦f)(x),

♦♥❞❡ (g_◦f)(x) =g f(x)

, ♣❛r❛ t♦❞♦ x_∈Df.

❖❜s❡r✈❛çã♦ ✶✳ ❙❛❧✈♦ s❡ ❤♦✉✈❡r ♠❡♥çã♦ ❡①♣❧✐❝✐t❛✱ t♦♠❛r❡♠♦s X = Df ❡

Y =R_. ❆ss✐♠✱ ✉s✉❛❧♠❡♥t❡ ❞❡♥♦t❛r❡♠♦s ✉♠❛ ❢✉♥çã♦ _f, ♥❛s ❢♦r♠❛s

f :Df ⊆R−→R, x7−→y=f(x),

♦✉ ♠❛✐s s✐♠♣❧❡s♠❡♥t❡

f :Df −→R, y =f(x).

✶✳✶ ❖♣❡r❛çõ❡s ❝♦♠ ❢✉♥çõ❡s

❉❡✜♥✐çã♦ ✽✳ ❙❡❥❛♠ ❢✉♥çõ❡s f, g :D_⊆R_→R_{, y =f(x)} ❡ _y=g(x)✳

✐✮ ❉❡✜♥✐♠♦s ❛ ❢✉♥çã♦ s♦♠❛ f +g :D _⊆R_→R_{, y = (f+g)(x),} ❝♦♠♦ ❛

❢✉♥çã♦(f +g)(x) = f(x) +g(x)✱ ♣❛r❛ t♦❞♦ x_∈D✳

✐✐✮ ❛ ❢✉♥çã♦ ♣r♦❞✉t♦ ♣♦r ✉♠ ❡s❝❛❧❛r(c_∈R_{)cf :D⊆}R_→R_{, y = (cf)(x),}

❝♦♠♦ ❛ ❢✉♥çã♦(cf)(x) =cf(x)✱ ♣❛r❛ t♦❞♦ x_∈D✳

✐✐✐✮ ❛ ❢✉♥çã♦ ♣r♦❞✉t♦ f g : D _⊆ R _→ R_{, y = (f g)(x),} ❝♦♠♦ ❛ ❢✉♥çã♦

(f g)(x) =f(x)g(x)✱ ♣❛r❛ t♦❞♦ x_∈D✳

✐✈✮ s❡ ❛ ❢✉♥çã♦ g(x) ₆= 0, ♣❛r❛ t♦❞♦ x _∈ D, ❡♥tã♦ ❛ ❢✉♥çã♦ q✉♦❝✐❡♥t❡ f

g :D⊆R→R, y =

f

g

(x), ♦♥❞❡ f g

(x) = f(x)

❈❛♣ít✉❧♦ ✷

❋✉♥çõ❡s ❆❧❣é❜r✐❝❛s

✷✳✶ ❋✉♥çõ❡s P♦❧✐♥♦♠✐❛✐s

❉❡✜♥✐çã♦ ✾✳ ❯♠❛ ❢✉♥çã♦ f é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ s❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣♦r

f(x) =anxn+. . .+a1x+a0,

♦♥❞❡ n é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ a0, a1, . . . , an, sã♦ ♥ú♠❡r♦s r❡❛✐s q✉❛✐sq✉❡r

❝♦♠ an 6= 0.

❯♠ ♥ú♠❡r♦ r❡❛❧ a é ❝❤❛♠❛❞♦ ❞❡ r❛í③ ❞❡f s❡ f(a) = 0. ◆❛t✉r❛❧♠❡♥t❡✱ t❡♠♦s q✉❡Df =R.

Pr♦♣♦s✐çã♦ ✸✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ a, ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ g✱ t❛❧ q✉❡

f(x)₋f(a) = (x₋a)g(x),

♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x.

✷✳✶✳✶ ❋✉♥çã♦ ❛✜♠

❉❡✜♥✐çã♦ ✶✵✳ ❯♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ f é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❛✜♠✱ s❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣♦r

f(x) =ax+b, ♦♥❞❡ a ❡ b sã♦ ♥ú♠❡r♦s r❡❛✐s q✉❛✐sq✉❡r✳

❖❜s❡r✈❛çã♦ ✷✳ ❯♠❛ ❢✉♥çã♦ ❛✜♠ f(x) =ax+b, ♥❛ q✉❛❧ a= 0, é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❝♦♥st❛♥t❡✳

✽ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

Pr♦♣♦s✐çã♦ ✹✳ ❙❡❥❛ ✉♠❛ ❢✉♥çã♦ ❛✜♠ f(x) = ax+b. ❊♥tã♦✿ ✐✮ s❡ a= 0, ❡♥tã♦ Im(f) =_{b_};

✐✐✮ s❡ a₆= 0, ❡♥tã♦ Im(f) =R_;

✐✐✮ s❡ a >0, ❡♥tã♦ f é ❝r❡s❝❡♥t❡ ❡♠ R_;

✐✐✐✮ s❡ a <0, ❡♥tã♦ f é ❞❡❝r❡s❝❡♥t❡ ❡♠ R_.

❋✐❣✉r❛ ✷✳✶✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡✱ ❢✭①✮❂❜✱ ♦♥❞❡ ❛❂♦✳

✷✳✷✳ ❋❯◆➬➹❖ ❱❆▲❖❘ ❆❇❙❖▲❯❚❖ ✾

❋✐❣✉r❛ ✷✳✸✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ ❛✜♠✱ ❢✭①✮❂❛①✰❜✱ ♦♥❞❡ ❛❁♦

✷✳✷ ❋✉♥çã♦ ❱❛❧♦r ❆❜s♦❧✉t♦

❉❡✜♥✐çã♦ ✶✶✳ ❈❤❛♠❛♠♦s ❛ ❢✉♥çã♦ f :R_→R✱ ❞❡✜♥✐❞❛ ♣♦r

f(x) =_|x|✱ ♣❛r❛ t♦❞♦ x_∈R✱ ❞❡ ❢✉♥çã♦ ✈❛❧♦r ❛❜s♦❧✉t♦✳

❆ ❢✉♥çã♦ ✈❛❧♦r ❛❜s♦❧✉t♦ é ✉♠❛ ❢✉♥çã♦ ♣❛r ❡ Im(f) = [0,+_∞[✳

✶✵ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

✷✳✷✳✶ ❋✉♥çã♦ ◗✉❛❞rát✐❝❛

❉❡✜♥✐çã♦ ✶✷✳ ❯♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ f é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ s❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣♦r ✉♠ ♣♦❧✐♥ô♠✐♦ q✉❛❞rát✐❝♦

f(x) =ax2 _+bx+c,

♦♥❞❡ a (₆= 0), b ❡ c sã♦ ♥ú♠❡r♦s r❡❛✐s q✉❛✐sq✉❡r✳

Pr♦♣♦s✐çã♦ ✺✳ ❙❡❥❛♠ ♥ú♠❡r♦s r❡❛✐s a (₆= 0), b ❡ c. ❊♥tã♦✱ ❛ ❧❡✐ ❞❡ ❝♦rr❡s✲ ♣♦♥❞ê♥❝✐❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ♣♦❞❡ ❡s❝r✐t❛ ♥❛ ❢♦r♠❛

f(x) = ax2_+bx+c=a

x+ b 2a

2

− b

2 _−4ac

4a ,

♣❛r❛ t♦❞♦ x_∈R_. ❆❧é♠ ❞✐ss♦✿

✐✮ s❡ a >0, ❡♥tã♦f ♣♦ss✉✐ ✈❛❧♦r ♠í♥✐♠♦ ❡♠x=₋ b

2a,❝♦♠ ✈❛❧♦r ✐♠❛❣❡♠

f ₋ b

2a

=₋b2

−4ac

4a ;

✐✐✮ s❡a <0, ❡♥tã♦f ♣♦ss✉✐ ✈❛❧♦r ♠á①✐♠♦ ❡♠ x=₋b

2a, ❝♦♠ ✈❛❧♦r ✐♠❛❣❡♠

f ₋ b

2a

=₋b2

−4ac

4a .

◆❡ss❡ ❝❛s♦✿

✐✐✐✮ s❡ a >0, ❡♥tã♦ Im(f) =

−b2

−4ac

4a ,+∞

;

✐✈✮ s❡ a <0, ❡♥tã♦ Im(f) =

− ∞,₋b2

−4ac

4a

;

✈✮ ♣❛r❛ t♦❞♦c_≥0, t❡♠♦s f ₋ b

2a −c

=f ₋ b

2a +c

.

Pr♦♣♦s✐çã♦ ✻✳ ❙❡❥❛♠ ♥ú♠❡r♦s r❡❛✐s a (₆= 0), b ❡ c t❛✐s q✉❡ b2

−4ac _≥ 0. ❊♥tã♦✱ ❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛ ❛❞♠✐t❡ ✉♠❛ ❞❡❝♦♠♣♦✲ s✐çã♦ ❞♦ t✐♣♦

f(x) = ax2_+bx+c

= a

"

x+ b 2a

2

−b

2_−4ac

4a2

#

= a

x₋−b+ √

b2_−4ac

2a x−

−b₋√b2_−4ac

2a

✷✳✷✳ ❋❯◆➬➹❖ ❱❆▲❖❘ ❆❇❙❖▲❯❚❖ ✶✶

♣❛r❛ t♦❞♦ x_∈R_. ❆ss✐♠✱ _{f(x) = 0} s❡✱ ❡ s♦♠❡♥t❡ s❡✱

x= −b+

√

b2 _−4ac

2a ♦✉ x=

−b₋√b2_−4ac

2a . ❆❧é♠ ❞✐ss♦✱ s❡ b2 _{−4ac < 0,} ❡♥tã♦✿

✐✮ a >0 ✐♠♣❧✐❝❛ ❡♠ f(x)>0, ♣❛r❛ t♦❞♦x_∈R_;

✐✐✮ a <0 ✐♠♣❧✐❝❛ ❡♠ f(x)<0, ♣❛r❛ t♦❞♦x_∈R_.

❋✐❣✉r❛ ✷✳✺✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ f(x) = ax2_{+bx+c, ♦♥❞❡ ❛❃♦}

✷✳✷✳✷ ❋✉♥çã♦ P♦tê♥❝✐❛

❉❡✜♥✐çã♦ ✶✸✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❯♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ f é ❝❤❛✲ ♠❛❞❛ ❞❡ ❢✉♥çã♦ ♣♦tê♥❝✐❛✱ s❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦

f(x) =xn_.

Pr♦♣♦s✐çã♦ ✼✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣❛r ♣♦s✐t✐✈♦✳ ❊♥tã♦✿

✐✮ ❛ ❢✉♥çã♦ ♣♦tê♥❝✐❛ é ✉♠❛ ❢✉♥çã♦ ♣❛r❀

✐✐✮ Im(f) = [0,+_∞[;

✶✷ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

❋✐❣✉r❛ ✷✳✻✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ q✉❛❞rát✐❝❛✱ f(x) = ax2_+bx+c, ♦♥❞❡ ❛❁♦

Pr♦♣♦s✐çã♦ ✽✳ ❙❡❥❛n ✉♠ ✐♥t❡✐r♦ ✐♠♣❛r ♣♦s✐t✐✈♦✳ ❊♥tã♦✿

✐✮ ❛ ❢✉♥çã♦ ♣♦tê♥❝✐❛ é ✉♠❛ ❢✉♥çã♦ ✐♠♣❛r❀

✐✐✮ Im(f) = R_;

✐✐✐✮ f é ❝r❡s❝❡♥t❡ ❡♠ R_.

✷✳✸✳ ❋❯◆➬➹❖ ❘❆❈■❖◆❆▲ ✶✸

❋✐❣✉r❛ ✷✳✽✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ ♣♦tê♥❝✐❛✱ f(x) = xn_{, ♦♥❞❡ ♥ é ✉♠ ✐♥t❡✐r♦}

✐♠♣❛r

✷✳✸ ❋✉♥çã♦ ❘❛❝✐♦♥❛❧

❉❡✜♥✐çã♦ ✶✹✳ ❯♠❛ ❢✉♥çã♦ f é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ r❛❝✐♦♥❛❧ s❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦

f(x) = p(x)

q(x),

♦♥❞❡ p=p(x) ❡ q =q(x) sã♦ ♣♦❧✐♥ô♠✐♦s ♥ã♦ ♥✉❧♦s q✉❡ ♥ã♦ tê♠ ❢❛t♦r❡s ❡♠

❝♦♠✉♠ ❡ ❝♦♠ ❣r❛✉ ❞❡ q ≥1.

❖ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ é ❞❛❞♦ ♣♦r Df ={x∈R |q(x)6= 0}.

❋✉♥çã♦ ❍✐♣ér❜♦❧❡

❉❡✜♥✐çã♦ ✶✺✳ ❯♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ ❤✐♣ér❜♦❧❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦

f(x) = ax+b

cx+d,

♦♥❞❡ a, b, c ❡ d sã♦ ♥ú♠❡r♦s r❡❛✐s ❝♦♠ c₆= 0 ❡ ad ₆=bc.

✶✹ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

✐✮ Df =R− {−

d c};

✐✐✮ Im(f) = R_{− {}a

c}.

❋✐❣✉r❛ ✷✳✾✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❤✐♣ér❜♦❧❡ f(x) = x−1

x+ 1

✷✳✸✳ ❋❯◆➬➹❖ ❘❆❈■❖◆❆▲ ✶✺

✷✳✸✳✶ ❙✉❣❡stõ❡s ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❣rá✜❝♦ ❞❡ ✉♠❛

❢✉♥çã♦ r❛❝✐♦♥❛❧

◆♦ ❝❛s♦ ❣❡r❛❧✱ ❞❡ ✉♠❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = p(x)/q(x), ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞♦ ❣rá✜❝♦✱ s✉❣❡r✐♠♦s ❞❡t❡r♠✐♥❛r ♦s s❡❣✉✐♥t❡s ✐t❡♥s✿

✐✮ ♦ ❞♦♠í♥✐♦ Df;

✐✐✮ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ✐♠❛❣❡♥s f(x), q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧ x t♦♠❛ ✈❛❧♦r❡s ♣ró①✐♠♦s ❞❛s r❛í③❡s ❞♦ ♣♦❧✐♥ô♠✐♦ q =q(x).

◆❡ss❡ ❝❛s♦✱ f(x)

=

p(x)

q(x) t♦♠❛ ✈❛❧♦r❡s ❝r❡s❝❡♥t❡s ❡ ✐❧✐♠✐t❛❞♦s✱

❛ ♠❡❞✐❞❛ q✉❡ ❛ ✈❛r✐á✈❡❧ x t♦♠❛ ✈❛❧♦r❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ♣ró①✐♠♦s ❞❡ q✉❛❧q✉❡r r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦q =q(x).❆❧é♠ ❞✐ss♦✱ q✉❛♥t♦ ♠❛✐s ♣ró①✐♠♦ ❢♦r ♦ ✈❛❧♦r q✉❡ x t♦♠❛ ❞❡ ✉♠❛ r❛✐③ ❞♦ ♣♦❧✐♥ô♠✐♦ q = q(x) ♠❛✐♦r é ♦

✈❛❧♦r ❞❡ f(x)

.

✐✐✐✮ ♦s ♣♦♥t♦s ❞❡ ❝♦rt❡s ♥♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s ox ❡ oy ✭s❡ ❡①✐st✐r❡♠✮❀ ✐✈✮ ❛♥á❧✐s❡ ❞♦ s✐♥❛❧ ❞❛s ✐♠❛❣❡♥s f(x), ♣❛r❛ x_∈Df;

✈✮ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛s ✐♠❛❣❡♥sf(x),q✉❛♥❞♦ ❛ ✈❛r✐á✈❡❧_|x_|t♦♠❛ ✈❛❧♦r❡s ❝r❡s❝❡♥t❡s✳

◆❡ss❡ ❝❛s♦✱ t❡♠♦s✿

✈✲✐✮ s❡ ❣r❛✉ ❞❡ p(x)<❣r❛✉ ❞❡q(x)✱ ❡♥tã♦ ❛ ✐♠❛❣❡♠ f(x) =p(x)

q(x)

t♦♠❛ ✈❛❧♦r❡s ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦ ❞❡ ③❡r♦✱ ❛ ♠❡❞✐❞❛ q✉❡ ❛ ✈❛✲ r✐á✈❡❧ |x_| t♦♠❛ ✈❛❧♦r❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s❀

✈✲✐✐✮ s❡ ❣r❛✉ ❞❡ p(x)_≥ ❣r❛✉ ❞❡ q(x)✱ ❡♥tã♦ ❡s❝r❡✈❛

p(x) =q(x)t(x) +r(x), ♦♥❞❡ ❣r❛✉ ❞❡r(x)< ❣r❛✉ ❞❡q(x), ❡ ❝♦♥❝❧✉❛ q✉❡

f(x) = p(x)

q(x) =t(x) +

r(x)

q(x)

t♦♠❛ ✈❛❧♦r❡s ❝❛❞❛ ✈❡③ ♠❛✐s ♣ró①✐♠♦ ❞❡t(x),❛ ♠❡❞✐❞❛ q✉❡ ❛ ✈❛r✐á✲ ✈❡❧ |x_| t♦♠❛ ✈❛❧♦r❡s s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡s✱ ❡♠ ❢✉♥çã♦ ❞❡ r(x) q(x)

✈❡r✐✜❝❛r ❛s ❝♦♥❞✐çõ❡s ❞♦ ✐t❡♠ ✈✲✐✮❀

✶✻ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

❋✐❣✉r❛ ✷✳✶✶✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = 1

x2 _{+x+ 1}

✷✳✹ ❋✉♥çã♦ ❘❛í③ ♥✲és✐♠❛

❚❡♦r❡♠❛ ✶✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣❛r ♣♦s✐t✐✈♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲♥❡❣❛t✐✈♦x,❡①✐st❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧ ♥ã♦✲♥❡❣❛t✐✈♦y, ❝❤❛♠❛❞♦ ❞❡ r❛✐③ n✲és✐♠❛ ❞❡ x, t❛❧ q✉❡ yn_=x.

❚❡♦r❡♠❛ ✷✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ✐♠♣❛r ♣♦s✐t✐✈♦✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧x, ❡①✐st❡ ✉♠ ú♥✐❝♦ ♥ú♠❡r♦ r❡❛❧ y, ❝❤❛♠❛❞♦ ❞❡ r❛✐③n✲és✐♠❛ ❞❡x, t❛❧ q✉❡ yn_=x.

◆♦t❛çã♦ ✸✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n, ❞❡♥♦t❛r❡♠♦s ❛s r❛í③❡sn✲és✐♠❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧x, ♦❜t✐❞♦s ♥♦s ❚❡♦r❡♠❛s ✶ ❡ ✷✱ ♣❡❧♦ sí♠❜♦❧♦

y= √n

x.

❈♦r♦❧ár✐♦ ✶✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣❛r ♣♦s✐t✐✈♦✳ ❊♥tã♦✿ ✐✮ √nxn=x, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ ♥ã♦ ♥❡❣❛t✐✈♦ x;

✐✐✮ √n

xn _=|x|, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ _x.

❈♦r♦❧ár✐♦ ✷✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ✐♠♣❛r ♣♦s✐t✐✈♦✳ ❊♥tã♦✿ ✐✮ √nxn=x, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x;

✐✐✮ √n

✷✳✹✳ ❋❯◆➬➹❖ ❘❆❮❩ ◆✲➱❙■▼❆ ✶✼

❋✐❣✉r❛ ✷✳✶✷✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = 5

−x2_{+ 4x−3}

❉❡✜♥✐çã♦ ✶✻✳ ❙❡❥❛ n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✳ ❯♠❛ ❢✉♥çã♦ f é ❝❤❛♠❛❞❛ ❞❡ ❢✉♥çã♦ r❛✐③ n✲és✐♠❛✱ s❡ s✉❛ ❧❡✐ ❞❡ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ é ❞❛❞❛ ♣❡❧❛ r❡❧❛çã♦

f(x) = √nx.

Pr♦♣♦s✐çã♦ ✶✵✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ ♣❛r ♣♦s✐t✐✈♦ n, ❛ ❢✉♥çã♦ r❛✐③ n✲és✐♠❛✱ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s✿

✐✮ Df = [0,+∞[;

✐✐✮ f é ❝r❡s❝❡♥t❡✱ ❡♠ t♦❞♦ ♦ s❡✉ ❞♦♠í♥✐♦ Df;

✐✐✐✮ Im(f) = [0,+_∞[.

Pr♦♣♦s✐çã♦ ✶✶✳ P❛r❛ t♦❞♦ ✐♥t❡✐r♦ ✐♠♣❛r ♣♦s✐t✐✈♦ n, ❛ ❢✉♥çã♦ r❛✐③ n✲és✐♠❛✱ s❛t✐s❢❛③ ❛s ♣r♦♣r✐❡❞❛❞❡s✿

✐✮ Df =R;

✐✐✮ f é ❝r❡s❝❡♥t❡✱ ❡♠ t♦❞♦ ♦ s❡✉ ❞♦♠í♥✐♦ Df;

✶✽ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

❋✐❣✉r❛ ✷✳✶✸✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = x+ 3

x2_{−6x+ 5}

❋✐❣✉r❛ ✷✳✶✹✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = x

2_{−6x+ 5}

✷✳✹✳ ❋❯◆➬➹❖ ❘❆❮❩ ◆✲➱❙■▼❆ ✶✾

❋✐❣✉r❛ ✷✳✶✺✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = x

2_−4x−5

x₋2

❋✐❣✉r❛ ✷✳✶✻✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ r❛❝✐♦♥❛❧ f(x) = x−2

✷✵ ❈❆P❮❚❯▲❖ ✷✳ ❋❯◆➬Õ❊❙ ❆▲●➱❇❘■❈❆❙

❋✐❣✉r❛ ✷✳✶✼✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ r❛✐③ n✲és✐♠❛✱ f(x) = √nx, ♦♥❞❡ ♥ é ✉♠

✐♥t❡✐r♦ ♣❛r

❋✐❣✉r❛ ✷✳✶✽✿ ●rá✜❝♦s ❞❛ ❢✉♥çã♦ r❛✐③ n✲és✐♠❛✱ f(x) = √nx, ♦♥❞❡ ♥ é ✉♠

❈❛♣ít✉❧♦ ✸

P♦tê♥❝✐❛ ❞❡ ✉♠ ♥ú♠❡r♦ r❡❛❧✳

❈♦♥t✐♥✉❛çã♦

✸✳✶ P♦tê♥❝✐❛s ❝♦♠ ❡①♣♦❡♥t❡s r❛❝✐♦♥❛✐s

❉❡✜♥✐çã♦ ✶✼✳ ❙❡❥❛ a (₆= 1) ✉♠ ♥ú♠❡r♦ r❡❛❧ q✉❛❧q✉❡r ♣♦s✐t✐✈♦ ❡ r = p

q,♦♥❞❡ p _∈ Z ❡ _{q ∈} N_, ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳ ❉❡✜♥✐♠♦s ❛ ♣♦tê♥❝✐❛ ❞❡ ❜❛s❡ _a ❡

❡①♣♦❡♥t❡ r ♦✉ s✐♠♣❧❡s♠❡♥t❡ ❛ ♣♦tê♥❝✐❛ ❞❡ a ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ r❡❛❧ ❞❛❞♦ ♣♦r

ar₌ √q

ap✳

Pr♦♣♦s✐çã♦ ✶✷✳ ❙❡❥❛ a ✉♠ ♥ú♠❡r♦ r❡❛❧ ❡ r, s ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✳ ❊♥tã♦✿ ✐✮ ar+s _=ar_as_;

✐✐✮ (ar₎s

=ars_;

✐✐✐✮ s❡ a >1, ❡♥tã♦ r < s ✐♠♣❧✐❝❛ ❡♠ ar _{< a}s

;

✐✈✮ s❡ 0< a <1, ❡♥tã♦ r < s ✐♠♣❧✐❝❛ ❡♠ ar_{> a}s

.

❈❛♣ít✉❧♦ ✹

❋✉♥çõ❡s ❊①♣♦♥ê♥❝✐❛✐s ❡

▲♦❣❛rít♠✐❝❛s

✹✳✶ ❋✉♥çõ❡s ❊①♣♦♥❡♥❝✐❛✐s✳ P♦tê♥❝✐❛s ❞❡ ❊①♣♦✲

❡♥t❡s ❘❡❛✐s

❚❡♦r❡♠❛ ✸✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ a (>0, ₆= 1), ❡①✐st❡ ✉♠❛ ú♥✐❝❛ ❢✉♥çã♦ ❜✐❥❡t♦r❛

fa:R→]0,+∞[, x7→y=fa(x),

❞❡♥♦♠✐♥❛❞❛ ❞❡ ❢✉♥çã♦ ❡①♣♦♥ê♥❝✐❛❧ ❞❡ ❜❛s❡ a, t❛❧ q✉❡

fa(r) =ar(=

q

√ ap_),

♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ r = p

q, ♦♥❞❡ p∈Z ❡ q ∈N.

◆❡st❡ ❝❛s♦✱ ❞❡♥♦t❛r❡♠♦s ♦s ✈❛❧♦r❡s ✐♠❛❣❡♥s ❞❛ ❢✉♥çã♦fa ♣❡❧♦ s✐♠❜♦❧♦

fa(x) = ax,

♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x.

❊♠ ♣❛rt✐❝✉❧❛r t❡♠♦s a0 _{= 1} ❡ _a1 _=a.

❚❡♦r❡♠❛ ✹✳ ❙❡❥❛ a (>0, ₆= 1) ♥ú♠❡r♦ r❡❛❧✳ ❊♥tã♦✱ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s

♥ú♠❡r♦s r❡❛✐s w ❡ x, t❡♠♦s✿

✷✹ ❈❆P❮❚❯▲❖ ✹✳ ❋❯◆➬Õ❊❙ ❊❳P❖◆✃◆❈■❆■❙ ❊ ▲❖●❆❘❮❚▼■❈❆❙

✐✮ aw+x _=aw_ax_;

✐✐✮ awx

=awx_;

✐✐✮ ❙❡ a >1, ❡♥tã♦ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ a é ❝r❡s❝❡♥t❡ ❡♠ R_;

✐✈✮ ❙❡ 0 < a < 1, ❡♥tã♦ ❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ ❜❛s❡ a é ❞❡❝r❡s❝❡♥t❡ ❡♠

R_.

●rá✜❝♦s ❞❛s ❢✉♥çõ❡s ❡①♣♦♥❡♥❝✐❛✐s ❞❡ ❜❛s❡a

❋✐❣✉r❛ ✹✳✶✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥ê♥❝✐❛❧✱ f(x) = ax_{✱ ♦♥❞❡ ❛❃✶}

✹✳✷✳ ❋❯◆➬➹❖ ▲❖●❆❘❮❚▼■❈❆ ✷✺

✹✳✷ ❋✉♥çã♦ ▲♦❣❛rít♠✐❝❛

❉❡✜♥✐çã♦ ✶✽✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ a (> 0, ₆= 1), ❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ a ❝♦♠♦ s❡♥❞♦ ❛ ❢✉♥çã♦ ✐♥✈❡rs❛ ❞❛ ❢✉♥çã♦ ❡①♣♦♥ê♥❝✐❛❧ ❞❡ ❜❛s❡ a fa. ❆ss✐♠✱ ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ a é ❛ ❢✉♥çã♦

f−1

a :]0,+∞[→R, x7→y=f−

1

a (x),

♦♥❞❡ y=f−1

a (x) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ fa(y) =x, ♣❛r❛ t♦❞♦ x∈]0,+∞[.

◆❡st❡ ❝❛s♦✱ ❞❡♥♦t❛r❡♠♦s ♦s ✈❛❧♦r❡s ✐♠❛❣❡♥s ❞❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ a, ♣❡❧♦ sí♠❜♦❧♦ f−1

a (x) = logax, ♣❛r❛ t♦❞♦x∈]0,+∞[. ❆ss✐♠✱

y= logax s❡✱ ❡ s♦♠❡♥t❡ s❡✱ay =x,♣❛r❛ t♦❞♦ x∈]0,+∞[.

❙❡❣✉❡ ❞✐ss♦ q✉❡✿

✐✮ alogax =x, ♣❛r❛ t♦❞♦ x∈]0,+∞[;

✐✐✮ logaay =y, ♣❛r❛ t♦❞♦y∈R.

❊♠ ♣❛rt✐❝✉❧❛r✱ t❡♠♦s q✉❡ log_a1 = 0 ❡ log_aa= 1.

Pr♦♣♦s✐çã♦ ✶✸✳ ❙❡❥❛♠a ❡b (>0,₆= 1)♥ú♠❡r♦s r❡❛✐s✳ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠

♦s ♥ú♠❡r♦s r❡❛✐s c_∈R ❡ _{w, x∈]0,+∞[,} t❡♠♦s q✉❡✿

✐✮ loga(wx) = logaw+ logax;

✐✐✮ log_axc _=clog ax;

✐✐✐✮ ❙❡ ❛ ❃ ✶✱ ❡♥tã♦ ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ ❛ é ❝r❡s❝❡♥t❡ ❡♠]0,+_∞[;

✐✈✮ ❙❡ ✵ ❁ ❛ ❁ ✶✱ ❡♥tã♦ ❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛ ❞❡ ❜❛s❡ ❛ é ❞❡❝r❡s❝❡♥t❡ ❡♠

]0,+_∞[. ❆❧é♠ ❞✐ss♦✱

✈✮ log_a(w

x) = logaw−logax;

✈✐✮ logax=

log_bx

logba

.

✷✻ ❈❆P❮❚❯▲❖ ✹✳ ❋❯◆➬Õ❊❙ ❊❳P❖◆✃◆❈■❆■❙ ❊ ▲❖●❆❘❮❚▼■❈❆❙

❋✐❣✉r❛ ✹✳✸✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❧♦❣❛rít♠✐❝❛✱ f(x) = logax✱ ♦♥❞❡ ❛❃✶

✹✳✸ ❖ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧

e

❈♦♥s✐❞❡r❡♠♦s ❛ s✉❝❡ssã♦ ❞❡ ♣♦tê♥❝✐❛s ❞❡ ♥ú♠❡r♦s r❡❛✐s✱ ❞❡✜♥✐❞❛ ♣♦r

1 + 1 1

1

,

1 + 1 2

2

,

1 + 1 3

3

,

1 + 1 4

4

,

1 + 1 5

5

, . . .

◗✉❛♥❞♦ n ❝r❡s❝❡ ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ❛ ❡①♣r❡ssã♦

1 + 1

n

n

s❡ ❛♣r♦①✐♠❛ ❞❡ ✉♠ ♥ú♠❡r♦ ✐rr❛❝✐♦♥❛❧ q✉❡ é r❡♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❧❡tr❛e, ❝✉❥♦ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ é2,7182. . . .

❆ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧ q✉❡ ❛♣❛r❡❝❡ ❝♦♠ ♠❛✐♦r ❢r❡qüê♥❝✐❛ ❡♠ ♠♦❞❡❧❛❣❡♠ ♠❛t❡♠át✐❝❛ é ❛ q✉❡ t❡♠ ♦ ♥ú♠❡r♦ e ❝♦♠♦ ❜❛s❡✳ ❈♦♠♦ e > 1, ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡①♣♦♥❡♥❝✐❛❧fe(x) =ex é s❡♠❡❧❤❛♥t❡ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦fa(x) =ax,

❝♦♠ a >1.

❉❡✜♥✐çã♦ ✶✾✳ ❉❛❞♦ ✉♠ ♥ú♠❡r♦ r❡❛❧ x >0, ❝❤❛♠❛r❡♠♦s ♦ ♥ú♠❡r♦ r❡❛❧ y, q✉❡ s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡

y= logex s❡✱ ❡ s♦♠❡♥t❡ s❡✱ y=f−

1

e (x)

s❡✱ ❡ s♦♠❡♥t❡ s❡✱ fe(y) =x s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ey =x,

❞❡ ❧♦❣❛r✐t♠♦ ♥❛t✉r❛❧ ❞❡ x, ❡ ♦ ❞❡♥♦t❛r❡♠♦s ♣❡❧♦ sí♠❜♦❧♦

✹✳✸✳ ❖ ◆Ú▼❊❘❖ ■❘❘❆❈■❖◆❆▲ E ✷✼

❈❛♣ít✉❧♦ ✺

❋✉♥çõ❡s ❚r✐❣♦♥♦♠étr✐❝❛s

✺✳✶ ❋✉♥çõ❡s ❙❡♥♦ ❡ ❈♦✲s❡♥♦

❚❡♦r❡♠❛ ✺✳ ❊①✐st❡ ✉♠ ú♥✐❝♦ ♣❛r ❞❡ ❢✉♥çõ❡s✱ ❞❡♥♦♠✐♥❛❞❛s ❞❡ ❢✉♥çõ❡s s❡♥♦ ❡ ❝♦✲s❡♥♦✱ ❞❡♥♦t❛❞❛s ♣♦r✿

sin : R_→R ❡ _{cos :}R_→R_,

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ✈❡r✐✜❝❛♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

❚✶✮ sin 0 = 0 ❡ cos 0 = 1❀

❚✷✮ ❆ ❢✉♥çã♦ s❡♥♦ é ✉♠❛ ❢✉♥çã♦ ✐♠♣❛r❀

❚✸✮ ❆ ❢✉♥çã♦ ❝♦✲s❡♥♦ é ✉♠❛ ❢✉♥çã♦ ♣❛r❀

❚✹✮ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s r❡❛✐s a ❡ b,

sin(a+b) = sinacosb+ sinbcosa❀

❚✺✮ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s r❡❛✐s a ❡ b,

cos(a+b) = cosacosb₋sinasinb❀

❚✻✮ ❊①✐st❡ ✉♠ ♠❡♥♦r ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✱ ❞❡♥♦t❡❞♦ ♣❡❧♦ s✐♠❜♦❧♦π, t❛❧ q✉❡

sin π 2

= 1 ❡ cos π 2) = 0❀

✸✵ ❈❆P❮❚❯▲❖ ✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙

❚✼✮ ❆ ❢✉♥çã♦ s❡♥♦ é ❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ [0,π

2]❀

❚✽✮ ❆ ❢✉♥çã♦ ❝♦✲s❡♥♦ é ❞❡❝r❡s❝❡♥t❡ ♥♦ ✐♥t❡r✈❛❧♦ [0,π

2].

▲❡♠❛ ✶✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x t❡♠♦s✿ ✐✮ sin2_{x+ cos}2_{x= 1;}

✐✐✮ −1_≤sinx_≤1 ❡ ₋1_≤cosx_≤1;

Pr♦♣♦s✐çã♦ ✶✹✳ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s r❡❛✐s a ❡ b, t❡♠♦s✿ ✐✮ sin(a₋b) = sinacosb₋sinbcosa;

✐✐✮ cos(a₋b) = cosacosb+ sinasinb.

❈♦r♦❧ár✐♦ ✸✳ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ♥ú♠❡r♦s r❡❛✐s a ❡ b, t❡♠♦s✿ ✐✮ 2 sinacosb= sin(a+b) + sin(a₋b);

✐✐✮ 2 sinbcosa= sin(a+b)₋sin(a₋b);

✐✐✐✮ 2 cosacosb= cos(a₋b) + cos(a+b);

✐✈✮ 2 sinasinb= cos(a₋b)₋cos(a+b). ▲❡♠❛ ✷✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x t❡♠♦s✿

✐✮ sin(2x) = 2 sinxcosx;

✐✐✮ sin2_x= 1

2− 1

2cos(2x);

✐✐✐✮ cos2_x= 1

2+ 1

2cos(2x).

✺✳✶✳✶ ■❞❡♥t✐❞❛❞❡s ❋✉♥❞❛♠❡♥t❛✐s

✐✮ sin x+ π 2

= cosx ❡ cos x+π 2

=₋sinx;

✐✐✮ sin x+π

=₋sinx ❡ cos x+π

✺✳✶✳ ❋❯◆➬Õ❊❙ ❙❊◆❖ ❊ ❈❖✲❙❊◆❖ ✸✶

✐✐✐✮ sin x+3π 2

=₋cosx ❡ cos x+ 3π 2

= sinx;

✐✈✮ sin x+ 2π

= sinx ❡ cos x+ 2π

= cosx.

Pr♦♣♦s✐çã♦ ✶✻✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x, t❡♠♦s✿

✐✮ sin x₋ π

2

=₋cosx ❡ cos x₋π

2

= sinx;

✐✐✮ sin x₋π

=₋sinx ❡ cos x₋π

=₋cosx;

✐✐✐✮ sin x₋ 3π

2

= cosx ❡ cos x₋3π

2

=₋sinx;

✐✈✮ sin x₋2π

= sinx ❡ cos x₋2π

= cosx.

❋✐❣✉r❛ ✺✳✶✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡♥♦✱ f(x) = sinx

✺✳✶✳✷ ❊q✉❛çõ❡s ❚r✐❣♦♥♦♠étr✐❝❛s

✐✮ sinx= 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x=kπ, ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k;

✐✐✮ sinx= 1 (sinx=₋1) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x= π

2 + 2kπ (x= 3π

2 + 2kπ),

✸✷ ❈❆P❮❚❯▲❖ ✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙

❋✐❣✉r❛ ✺✳✷✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦✲s❡♥♦✱ f(x) = cosx

✐✐✐✮ cosx= 0 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x= 2k+ 1 2

π, ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k;

✐✈✮ cosx= 1 (cosx=₋1)s❡✱ ❡ s♦♠❡♥t❡ s❡✱ x= 2kπ (x= (2k+ 1)π),♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k.

▲❡♠❛ ✹✳ ❆s r❡❧❛çõ❡s ✈❛❧❡♠✱ ❛❜❛✐①♦✿

✐✮ sina= sinb s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b+ 2kπ ♦✉ a=₋b+ (2k+ 1)π, ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k;

✐✐✮ cosa= cosb s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b+ 2kπ ♦✉ a=₋b+ 2kπ, ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k;

✐✐✐✮ sina= cosb s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a=b+ 4k+ 1

2

π ♦✉ a=₋b+ 4k+ 1 2

π, ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ k.

✺✳✷ ❋✉♥çõ❡s ❚❛♥❣❡♥t❡ ❡ ❈♦✲t❛♥❣❡♥t❡

❉❡✜♥✐çã♦ ✷✵✳ ❉❡✜♥✐♠♦s ❛ ❢✉♥çã♦ t❛♥❣❡♥t❡ ❡ ❝♦✲t❛♥❣❡♥t❡✱ ❞❡♥♦t❛❞❛s ♣❡❧♦s s✐♠❜♦❧♦s

✺✳✷✳ ❋❯◆➬Õ❊❙ ❚❆◆●❊◆❚❊ ❊ ❈❖✲❚❆◆●❊◆❚❊ ✸✸

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠♦ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✉❥❛s r❡❣r❛s ❞❡ ❝♦rr❡s♣♦♥✲ ❞ê♥❝✐❛s ❡ s❡✉s ❞♦♠í♥✐♦s sã♦ ❞❛❞♦s ♣♦r

tanx= sinx

cosx, ♣❛r❛ t♦❞♦ x∈Dtan, ♦♥❞❡ Dtan ={x∈R | cosx6= 0} ❡

cotx= cosx

sinx, ♣❛r❛ t♦❞♦ x∈Dcot, ♦♥❞❡ Dcot ={x∈R | sinx6= 0}.

Pr♦♣♦s✐çã♦ ✶✼✳ ❆s r❡❧❛çõ❡s ✈❛❧❡♠✱ ❛❜❛✐①♦✿

✐✮ tan(₋x) = ₋tanx, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x_∈Dtan;

✐✐✮ cot(₋x) = ₋cotx, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x_∈Dcot.

Pr♦♣♦s✐çã♦ ✶✽✳ P❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x_∈Dtan, t❡♠♦s✿

✐✮ tan x+π 2

=₋cotx ❡ tan x+π

= tanx;

✐✐✮ tan x₋ π

2

=₋cotx ❡ tan x₋π

= tanx.

✸✹ ❈❆P❮❚❯▲❖ ✺✳ ❋❯◆➬Õ❊❙ ❚❘■●❖◆❖▼➱❚❘■❈❆❙

❋✐❣✉r❛ ✺✳✹✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ❝♦✲t❛♥❣❡♥t❡✱ f(x) = cotx

✺✳✸ ❋✉♥çõ❡s ❙❡❝❛♥t❡ ❡ ❈♦✲s❡❝❛♥t❡

❉❡✜♥✐çã♦ ✷✶✳ ❉❡✜♥✐♠♦s ❛ ❢✉♥çã♦ s❡❝❛♥t❡ ❡ ❝♦✲s❡❝❛♥t❡✱ ❞❡♥♦t❛❞❛s ♣❡❧♦s s✐♠❜♦❧♦s

sec :Dsec ⊂R→R ❡ csc :Dcsc ⊂R→R,

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❝♦♠♦ ❛s ❢✉♥çõ❡s tr✐❣♦♥♦♠étr✐❝❛s ❝✉❥❛s r❡❣r❛s ❞❡ ❝♦rr❡s♣♦♥✲ ❞ê♥❝✐❛s ❡ s❡✉s ❞♦♠í♥✐♦s sã♦ ❞❛❞♦s ♣♦r

secx= 1

cosx, ♣❛r❛ t♦❞♦x∈Dsec, ♦♥❞❡ Dsec ={x∈R | cosx6= 0} (= Dtan) ❡

cscx= 1

sinx, ♣❛r❛ t♦❞♦x∈Dcsc, ♦♥❞❡ Dcsc ={x∈R | sinx6= 0} (= Dcot).

Pr♦♣♦s✐çã♦ ✶✾✳ ❆s r❡❧❛çõ❡s ✈❛❧❡♠✱ ❛❜❛✐①♦✿

✐✮ 1 + tan2_{x= sec}2_{x, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x∈D}

sec (= Dtan);

✐✐✮ 1 + cot2_{x= csc}2_{x, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x∈D}

csc (= Dcot).

✺✳✸✳ ❋❯◆➬Õ❊❙ ❙❊❈❆◆❚❊ ❊ ❈❖✲❙❊❈❆◆❚❊ ✸✺

✐✮ secx_{≤ −}1 ♦✉ secx_≥1, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x_∈Dsec;

✐✐✮ cscx_{≤ −}1 ♦✉ cscx_≥1, ♣❛r❛ t♦❞♦ ♥ú♠❡r♦ r❡❛❧ x_∈Dcsc.

❋✐❣✉r❛ ✺✳✺✿ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ s❡❝❛♥t❡✱ f(x) = secx