有限数学示例

f (x) = \frac{x}{\sqrt[3]{x^{2} - 1}}

$f (x) = \frac{x}{\sqrt[3]{x^{2} - 1}}$

解题步骤 1

化简

f (x)

$f (x)$ 。

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解题步骤 1.1

化简分母。

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解题步骤 1.1.1

将

1

$1$ 重写为

1^{2}

$1^{2}$ 。

f (x) = \frac{x}{\sqrt[3]{x^{2} - 1^{2}}}

$f (x) = \frac{x}{\sqrt[3]{x^{2} - 1^{2}}}$

解题步骤 1.1.2

因为两项都是完全平方数，所以使用平方差公式

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ 进行因式分解，其中

a = x

$a = x$ 和

b = 1

$b = 1$ 。

f (x) = \frac{x}{\sqrt[3]{(x + 1) (x - 1)}}

$f (x) = \frac{x}{\sqrt[3]{(x + 1) (x - 1)}}$

f (x) = \frac{x}{\sqrt[3]{(x + 1) (x - 1)}}

$f (x) = \frac{x}{\sqrt[3]{(x + 1) (x - 1)}}$

解题步骤 1.2

将

\frac{x}{\sqrt[3]{(x + 1) (x - 1)}}

$\frac{x}{\sqrt[3]{(x + 1) (x - 1)}}$ 乘以

\frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}

$\frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}$ 。

f (x) = \frac{x}{\sqrt[3]{(x + 1) (x - 1)}} \cdot \frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}

$f (x) = \frac{x}{\sqrt[3]{(x + 1) (x - 1)}} \cdot \frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}$

解题步骤 1.3

合并和化简分母。

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解题步骤 1.3.1

将

\frac{x}{\sqrt[3]{(x + 1) (x - 1)}}

$\frac{x}{\sqrt[3]{(x + 1) (x - 1)}}$ 乘以

\frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}

$\frac{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{2}}$ 。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{\sqrt[3]{(x + 1) (x - 1)} {\sqrt[3]{(x + 1) (x - 1)}}^{2}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{\sqrt[3]{(x + 1) (x - 1)} {\sqrt[3]{(x + 1) (x - 1)}}^{2}}$

解题步骤 1.3.2

对

\sqrt[3]{(x + 1) (x - 1)}

$\sqrt[3]{(x + 1) (x - 1)}$ 进行

1

$1$ 次方运算。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{\sqrt[3]{(x + 1) (x - 1)} {\sqrt[3]{(x + 1) (x - 1)}}^{2}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{\sqrt[3]{(x + 1) (x - 1)} {\sqrt[3]{(x + 1) (x - 1)}}^{2}}$

解题步骤 1.3.3

使用幂法则

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 合并指数。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{1 + 2}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{1 + 2}}$

解题步骤 1.3.4

将

1

$1$ 和

2

$2$ 相加。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{3}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{\sqrt[3]{(x + 1) (x - 1)}}^{3}}$

解题步骤 1.3.5

将

{\sqrt[3]{(x + 1) (x - 1)}}^{3}

${\sqrt[3]{(x + 1) (x - 1)}}^{3}$ 重写为

(x + 1) (x - 1)

$(x + 1) (x - 1)$ 。

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解题步骤 1.3.5.1

使用

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ ，将

\sqrt[3]{(x + 1) (x - 1)}

$\sqrt[3]{(x + 1) (x - 1)}$ 重写成

{((x + 1) (x - 1))}^{\frac{1}{3}}

${((x + 1) (x - 1))}^{\frac{1}{3}}$ 。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{({((x + 1) (x - 1))}^{\frac{1}{3}})}^{3}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{({((x + 1) (x - 1))}^{\frac{1}{3}})}^{3}}$

解题步骤 1.3.5.2

运用幂法则并将指数相乘，

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ 。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{1}{3} \cdot 3}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{1}{3} \cdot 3}}$

解题步骤 1.3.5.3

组合

\frac{1}{3}

$\frac{1}{3}$ 和

3

$3$ 。

f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{3}{3}}}

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{3}{3}}}$

解题步骤 1.3.5.4

约去

3

$3$ 的公因数。

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解题步骤 1.3.5.4.1

约去公因数。

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{{((x + 1) (x - 1))}^{\frac{3}{3}}}$

解题步骤 1.3.5.4.2

重写表达式。

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{(x + 1) (x - 1)}$

解题步骤 1.3.5.5

化简。

$f (x) = \frac{x {\sqrt[3]{(x + 1) (x - 1)}}^{2}}{(x + 1) (x - 1)}$

解题步骤 1.4

化简分子。

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解题步骤 1.4.1

将

${\sqrt[3]{(x + 1) (x - 1)}}^{2}$ 重写为

$\sqrt[3]{{((x + 1) (x - 1))}^{2}}$ 。

$f (x) = \frac{x \sqrt[3]{{((x + 1) (x - 1))}^{2}}}{(x + 1) (x - 1)}$

解题步骤 1.4.2

对

$(x + 1) (x - 1)$ 运用乘积法则。

$f (x) = \frac{x \sqrt[3]{{(x + 1)}^{2} {(x - 1)}^{2}}}{(x + 1) (x - 1)}$

解题步骤 2

The word linear is used for a straight line. A linear function is a function of a straight line, which means that the degree of a linear function must be

$0$ or

$1$ . In this case, The degree of

$f (x) = \frac{x \sqrt[3]{{(x + 1)}^{2} {(x - 1)}^{2}}}{(x + 1) (x - 1)}$ is

$- 1$ , which makes the function a nonlinear function.

$f (x) = \frac{x \sqrt[3]{{(x + 1)}^{2} {(x - 1)}^{2}}}{(x + 1) (x - 1)}$ is not a linear function

有限数学 示例

有限数学示例