Finite Mathematik Beispiele

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

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

Schritt 1

Vereinfache

f (x)

$f (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1

Vereinfache den Nenner.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.1.1

Schreibe

1

$1$ als

1^{2}

$1^{2}$ um.

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

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

Schritt 1.1.2

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel,

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

$a^{2} - b^{2} = (a + b) (a - b)$ , mit

a = x

$a = x$ und

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)}}

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

Schritt 1.2

Mutltipliziere

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

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

\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}}

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

Schritt 1.3

Vereinige und vereinfache den Nenner.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.3.1

Mutltipliziere

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

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

\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}}

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

Schritt 1.3.2

Potenziere

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

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

1

$1$ .

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

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

Schritt 1.3.3

Wende die Exponentenregel

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

$a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

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

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

Schritt 1.3.4

Addiere

1

$1$ und

2

$2$ .

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

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

Schritt 1.3.5

Schreibe

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

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

(x + 1) (x - 1)

$(x + 1) (x - 1)$ um.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.3.5.1

Benutze

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

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

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

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

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

${((x + 1) (x - 1))}^{\frac{1}{3}}$ neu zu schreiben.

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

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

Schritt 1.3.5.2

Wende die Potenzregel an und multipliziere die Exponenten,

{(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}}

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

Schritt 1.3.5.3

Kombiniere

\frac{1}{3}

$\frac{1}{3}$ und

3

$3$ .

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

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

Schritt 1.3.5.4

Kürze den gemeinsamen Faktor von

3

$3$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 1.3.5.4.1

Kürze den gemeinsamen Faktor.

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

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

Schritt 1.3.5.4.2

Forme den Ausdruck um.

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

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

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

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

Schritt 1.3.5.5

Vereinfache.

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

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

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

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

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

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

Schritt 1.4

Vereinfache den Zähler.

Tippen, um mehr Schritte zu sehen ...

Schritt 1.4.1

Schreibe

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

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

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

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

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

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

Schritt 1.4.2

Wende die Produktregel auf

(x + 1) (x - 1)

$(x + 1) (x - 1)$ an.

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

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

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

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

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

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

Schritt 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

$0$ or

1

$1$ . In this case, The degree of

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

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

- 1

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

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

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