Calculus Examples

$\sqrt{1 - 3^{t}}$

Step 1

Set the radicand in $\sqrt{1 - 3^{t}}$ greater than or equal to $0$ to find where the expression is defined.

$1 - 3^{t} \geq 0$

Step 2

Solve for $t$ .

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Step 2.1

Subtract $1$ from both sides of the inequality.

$- 3^{t} \geq - 1$

Step 2.2

Divide each term in $- 3^{t} \geq - 1$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- 3^{t} \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3^{t}}{- 1} \leq \frac{- 1}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{3^{t}}{1} \leq \frac{- 1}{- 1}$

Step 2.2.2.2

Divide $3^{t}$ by $1$ .

$3^{t} \leq \frac{- 1}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 1$ by $- 1$ .

$3^{t} \leq 1$

Step 2.3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (3^{t}) \leq \ln (1)$

Step 2.4

Expand $\ln (3^{t})$ by moving $t$ outside the logarithm.

$t \ln (3) \leq \ln (1)$

Step 2.5

Simplify the right side.

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Step 2.5.1

The natural logarithm of $1$ is $0$ .

$t \ln (3) \leq 0$

Step 2.6

Divide each term in $t \ln (3) \leq 0$ by $\ln (3)$ and simplify.

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Step 2.6.1

Divide each term in $t \ln (3) \leq 0$ by $\ln (3)$ .

$\frac{t \ln (3)}{\ln (3)} \leq \frac{0}{\ln (3)}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $\ln (3)$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{t \ln (3)}{\ln (3)} \leq \frac{0}{\ln (3)}$

Step 2.6.2.1.2

Divide $t$ by $1$ .

$t \leq \frac{0}{\ln (3)}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Divide $0$ by $\ln (3)$ .

$t \leq 0$

Step 3

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

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${t | t \leq 0}$

Step 4