Calculus Examples

$f (x) = 750 e^{- 1.5 x} - 1125 e^{- 1.5} x$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Change $- 1.5$ into a fraction.

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Step 1.1.1

Multiply by $\frac{10}{10}$ to remove the decimal.

$750 e^{- 1.5 x} - 1125 e^{\frac{10 \cdot - 1.5}{10}} x$

Step 1.1.2

Multiply $10$ by $- 1.5$ .

$750 e^{- 1.5 x} - 1125 e^{\frac{- 15}{10}} x$

Step 1.1.3

Move the negative in front of the fraction.

$750 e^{- 1.5 x} - 1125 e^{- \frac{15}{10}} x$

Step 1.1.4

Cancel the common factor of $15$ and $10$ .

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Step 1.1.4.1

Factor $5$ out of $15$ .

$750 e^{- 1.5 x} - 1125 e^{- \frac{5 (3)}{10}} x$

Step 1.1.4.2

Cancel the common factors.

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Step 1.1.4.2.1

Factor $5$ out of $10$ .

$750 e^{- 1.5 x} - 1125 e^{- \frac{5 \cdot 3}{5 \cdot 2}} x$

Step 1.1.4.2.2

Cancel the common factor.

$750 e^{- 1.5 x} - 1125 e^{- \frac{5 \cdot 3}{5 \cdot 2}} x$

Step 1.1.4.2.3

Rewrite the expression.

$750 e^{- 1.5 x} - 1125 e^{- \frac{3}{2}} x$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$750 e^{- 1.5 x} - 1125 \frac{1}{e^{\frac{3}{2}}} x$

Step 1.3

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$750 e^{- 1.5 x} - 1125 \frac{1}{\sqrt{e^{3}}} x$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

The range is all the valid $y$ values.

All real numbers

Step 4

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: All real numbers

Step 5