Calculus Examples

$f (t) = 1 - \frac{10}{t + 10} + \frac{100}{{(t + 10)}^{2}}$

Step 1

Set the denominator in $\frac{10}{t + 10}$ equal to $0$ to find where the expression is undefined.

$t + 10 = 0$

Step 2

Subtract $10$ from both sides of the equation.

$t = - 10$

Step 3

Set the denominator in $\frac{100}{{(t + 10)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(t + 10)}^{2} = 0$

Step 4

Solve for $t$ .

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Step 4.1

Set the $t + 10$ equal to $0$ .

$t + 10 = 0$

Step 4.2

Subtract $10$ from both sides of the equation.

$t = - 10$

Step 5

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

Interval Notation:

$(- \infty, - 10) \cup (- 10, \infty)$

Set-Builder Notation:

${t | t \neq - 10}$

Step 6

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[\frac{3}{4}, \infty)$

Set-Builder Notation:

${y | y \geq \frac{3}{4}}$

Step 7

Determine the domain and range.

Domain: $(- \infty, - 10) \cup (- 10, \infty), {t | t \neq - 10}$

Range: $[\frac{3}{4}, \infty), {y | y \geq \frac{3}{4}}$

Step 8