Precalculus Examples

If $f (x) = 3 - 2 x$ and $g (x) = \frac{1}{x + 5}$ ; what is the value of $(\frac{f}{g}) (8)$ ?

Step 1

Evaluate $\frac{f}{g}$ .

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Step 1.1

Replace the function designators in $\frac{f}{g}$ with the actual functions.

$\frac{3 - 2 x}{\frac{1}{x + 5}}$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$(3 - 2 x) (x + 5)$

Step 1.3

Expand $(3 - 2 x) (x + 5)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$3 (x + 5) - 2 x (x + 5)$

Step 1.3.2

Apply the distributive property.

$3 x + 3 \cdot 5 - 2 x (x + 5)$

Step 1.3.3

Apply the distributive property.

$3 x + 3 \cdot 5 - 2 x \cdot x - 2 x \cdot 5$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $3$ by $5$ .

$3 x + 15 - 2 x \cdot x - 2 x \cdot 5$

Step 1.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.2.1

Move $x$ .

$3 x + 15 - 2 (x \cdot x) - 2 x \cdot 5$

Step 1.4.1.2.2

Multiply $x$ by $x$ .

$3 x + 15 - 2 x^{2} - 2 x \cdot 5$

Step 1.4.1.3

Multiply $5$ by $- 2$ .

$3 x + 15 - 2 x^{2} - 10 x$

Step 1.4.2

Subtract $10 x$ from $3 x$ .

$- 7 x + 15 - 2 x^{2}$

Step 1.5

Move $15$ .

$- 7 x - 2 x^{2} + 15$

Step 1.6

Reorder $- 7 x$ and $- 2 x^{2}$ .

$- 2 x^{2} - 7 x + 15$

Step 2

Evaluate $\frac{f}{g}$ at $8$ .

$- 2 {(8)}^{2} - 7 \cdot 8 + 15$

Step 3

Simplify $- 2 {(8)}^{2} - 7 \cdot 8 + 15$ .

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Step 3.1

Remove parentheses.

$- 2 {(8)}^{2} - 7 \cdot 8 + 15$

Step 3.2

Simplify each term.

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Step 3.2.1

Raise $8$ to the power of $2$ .

$- 2 \cdot 64 - 7 \cdot 8 + 15$

Step 3.2.2

Multiply $- 2$ by $64$ .

$- 128 - 7 \cdot 8 + 15$

Step 3.2.3

Multiply $- 7$ by $8$ .

$- 128 - 56 + 15$

Step 3.3

Simplify by adding and subtracting.

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Step 3.3.1

Subtract $56$ from $- 128$ .

$- 184 + 15$

Step 3.3.2

Add $- 184$ and $15$ .

$- 169$