Question

1) Determine os coeficientes e as raízes das funções abaixo:
a) g(x) = 5 x - 1
b) y = x/5 + 3
C) f (x) = 3x
d) f (x) = - x - 4
2) Calcule o valor das funções anteriores para x = 1 e x = - 1.

Answer 1

Identificado os coeficientes, calculado as raízes de cada função e calculado o valor das funções, obtivemos:

>>> Questão 1:

• Item A: Coeficientes a=5, b= -1, e a raíz da função é 1/5;
• Item B: Coeficientes a= 1/5, b= 3. e a raíz da função é -15;
• Item C: Coeficiente a=3, b= 0, e a raíz da função é 0;
• Item D: Coeficiente a= -1, b= -4, e a raíz da função é -4.

>>> Questão 2:

• Item A: g(1)=4 e g(-1)= -6;
• Item B: y= 16/5 e y = 14/5;
• Item C: f(1)= 3 e f(-1)= -3;
• Item D: f(1)= -5 e f(-1)= -3.

Acompanhe a solução:

A função padrão é f(x)=ax + b, o qual a = coeficiente angular e b = coeficeinte linear.

Raíz da função ou Zero da função refere-se ao ponto que a reta corta o eixo x. Para isto, y = f(x) = 0.

Identificando os coeficientes:

>>> Item A:

$\large\begin {array}{l}g(x)=\underbrace{5}_{a}x\underbrace{-1}_{b}\\\\\Large\boxed{\boxed{a=5\;e\;b=-1}}\Huge\checkmark\end {array}$

>>> Item B:

$\large\begin {array}{l}y=\dfrac{x}{5}+3\\\\y=\underbrace{\dfrac{1}{5}}_{a}x\underbrace{+3}_{b}\\\\\Large\boxed{\boxed{a=\dfrac{1}{5}\;e\;b=3}}\Huge\checkmark\end {array}$

>>> Item C:

$\large\begin {array}{l}f(x)=\underbrace{3}_{a}x\\\\\Large\boxed{\boxed{a=3\;e\;b=0}}\Huge\checkmark\end {array}$

>>> Item D:

$\large\begin {array}{l}f((x)=-x-4\\\\f(x)=\underbrace{-1}_{a}x\underbrace{-4}_{b}\\\\\Large\boxed{\boxed{a=-1\;e\;b=-4}}\Huge\checkmark\end {array}$

Cálculo da raíz:

>>> Item A:

$\large\begin {array}{l}g(x)=5x-1\\\\0=5x-1\\\\1=5x\\\\\Large\boxed{\boxed{x=\dfrac{1}{5}}}\Huge\checkmark\end {array}$

>>> Item B:

$\large\begin {array}{l}y=\dfrac{x}{5}+3\\\\0=\dfrac{x}{5}+3\\\\\dfrac{x}{5}=-3\\\\x=-3\cdot5\\\\\Large\boxed{\boxed{x=-15}}\Huge\checkmark\end {array}$

>>> Item C:

$\large\begin {array}{l}f(x)=3x\\\\0=3x\\\\\Large\boxed{\boxed{x=0}}\Huge\checkmark\end {array}$

>>> Item D:

$\large\begin {array}{l}f((x)=-x-4\\\\0=-x-4\\\\-x=4\\\\\Large\boxed{\boxed{x=-4}}\Huge\checkmark\end {array}$

Cálculo do valor das funções quando x=1 e x = -1:

>>> Item A:

$\boxed{\large\begin {array}{l}>>>x=1:\\\\g(x)=5x-1\\\\g(1)=5\cdot1-1\\\\g(1)=5-1\\\\\Large\boxed{\boxed{g(1)=~~4~}}\Huge\checkmark\end {array}} \quad\quad \boxed{\large\begin {array}{l}>>>x=-1:\\\\g(x)=5x-1\\\\g(-1)=5\cdot(-1)-1\\\\g(-1)=-5-1\\\\\Large\boxed{\boxed{g(-1)=-6}}\Huge\checkmark\end {array}}$

>>> Item B:

$\boxed{\large\begin {array}{l}>>>x=1:\\\\y=\dfrac{x}{5}+3\\\\y=\dfrac{1}{5}+3\\\\y=\dfrac{1+15}{5}\\\\\Large\boxed{\boxed{y=~~\dfrac{16}{5}~}}\Huge\checkmark\end {array}} \quad\quad \boxed{\large\begin {array}{l}>>>x=-1:\\\\y=\dfrac{x}{5}+3\\\\y=-\dfrac{1}{5}+3\\\\y=\dfrac{-1+15}{5}\\\\\Large\boxed{\boxed{y=~~\dfrac{14}{5}~}}\Huge\checkmark\end {array}}$

>>> Item C:

$\boxed{\large\begin {array}{l}>>>x=1:\\\\f(x)=3x\\\\f(1)=3\cdot1\\\\\Large\boxed{\boxed{f(1)=~~3~}}\Huge\checkmark\end {array}} \quad\quad \boxed{\large\begin {array}{l}>>>x=-1:\\\\f(x)=3x\\\\f(-1)=3\cdot(-1)\\\\\Large\boxed{\boxed{f(-1)=-3}}\Huge\checkmark\end {array}}$

>>> Item D:

$\boxed{\large\begin {array}{l}>>>x=1:\\\\f(x)=-x-4\\\\f(1)=-1-4\\\\\Large\boxed{\boxed{f(1)=-5}}\Huge\checkmark\end {array}} \quad\quad \boxed{\large\begin {array}{l}>>>x=-1:\\\\f(x)=-x-4\\\\f(-1)=-(-1)-4\\\\\Large\boxed{\boxed{f(-1)=-3}}\Huge\checkmark\end {array}}$

Resposta:

Questão 1:

• Item A: Coeficientes a=5, b= -1, e a raíz da função é 1/5;
• Item B: Coeficientes a= 1/5, b= 3. e a raíz da função é -15;
• Item C: Coeficiente a=3, b= 0, e a raíz da função é 0;
• Item D: Coeficiente a= -1, b= -4, e a raíz da função é -4.

Questão 2:

• Item A: g(1)=4 e g(-1)= -6;
• Item B: y= 16/5 e y = 14/5;
• Item C: f(1)= 3 e f(-1)= -3;
• Item D: f(1)= -5 e f(-1)= -3.

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Bons estudos!