Calculus Examples

$h (x) = {\begin{cases} - x^{2} + k^{2}, 0 \leq x \leq 1 \\ \frac{4 x + 4}{2 - x}, x > 1 \end{cases}$

Step 1

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Step 1.1

To find whether the function is continuous on $(1, \infty)$ or not, find the domain of $f (x) = \frac{4 x + 4}{2 - x}$ .

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Step 1.1.1

Set the denominator in $\frac{4 x + 4}{2 - x}$ equal to $0$ to find where the expression is undefined.

$2 - x = 0$

Step 1.1.2

Solve for $x$ .

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Step 1.1.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 1.1.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 1.1.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 1.1.2.2.2

Simplify the left side.

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Step 1.1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 1.1.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 1.1.2.2.3

Simplify the right side.

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Step 1.1.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 1.1.3

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

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Step 1.2

$f (x)$ is not continuous on $(1, \infty)$ because $2$ is not in the domain of $f (x) = \frac{4 x + 4}{2 - x}$ .

The function is not continuous.

Step 2

Not continuous

Step 3