Finite Math Examples

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

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{3 x + 1}{x - x^{2}}$ equal to $0$ to find where the expression is undefined.

$x - x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Factor the left side of the equation.

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Step 1.2.1.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u - u^{2} = 0$

Step 1.2.1.2

Factor $u$ out of $u - u^{2}$ .

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Step 1.2.1.2.1

Raise $u$ to the power of $1$ .

$u - u^{2} = 0$

Step 1.2.1.2.2

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - u^{2} = 0$

Step 1.2.1.2.3

Factor $u$ out of $- u^{2}$ .

$u \cdot 1 + u (- u) = 0$

Step 1.2.1.2.4

Factor $u$ out of $u \cdot 1 + u (- u)$ .

$u (1 - u) = 0$

Step 1.2.1.3

Replace all occurrences of $u$ with $x$ .

$x (1 - x) = 0$

Step 1.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - x = 0$

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

Set $1 - x$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $1 - x$ equal to $0$ .

$1 - x = 0$

Step 1.2.4.2

Solve $1 - x = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 1.2.4.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 1.2.4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 1.2.5

The final solution is all the values that make $x (1 - x) = 0$ true.

$x = 0, 1$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 0, 1}$

Interval Notation:

$(- \infty, 0) \cup (0, 1) \cup (1, \infty)$

Set-Builder Notation:

${x | x \neq 0, 1}$

Step 2

Since the domain is not all real numbers, $\frac{3 x + 1}{x - x^{2}}$ is not continuous over all real numbers.

Not continuous

Step 3