Precalculus Examples

$f (x) = - x^{2} + 6 x - 1$ and $g (x) = 3 x^{2} - 4 x - 1$ ; find $(f - g) (x)$

Step 1

Evaluate $f - g$ .

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Step 1.1

Replace the function designators in $f - g$ with the actual functions.

$(- x^{2} + 6 x - 1) - (3 x^{2} - 4 x - 1)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$- x^{2} + 6 x - 1 - (3 x^{2}) - (- 4 x) - - 1$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Multiply $3$ by $- 1$ .

$- x^{2} + 6 x - 1 - 3 x^{2} - (- 4 x) - - 1$

Step 1.2.2.2

Multiply $- 4$ by $- 1$ .

$- x^{2} + 6 x - 1 - 3 x^{2} + 4 x - - 1$

Step 1.2.2.3

Multiply $- 1$ by $- 1$ .

$- x^{2} + 6 x - 1 - 3 x^{2} + 4 x + 1$

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Combine the opposite terms in $- x^{2} + 6 x - 1 - 3 x^{2} + 4 x + 1$ .

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Step 1.3.1.1

Add $- 1$ and $1$ .

$- x^{2} + 6 x - 3 x^{2} + 4 x + 0$

Step 1.3.1.2

Add $- x^{2} + 6 x - 3 x^{2} + 4 x$ and $0$ .

$- x^{2} + 6 x - 3 x^{2} + 4 x$

Step 1.3.2

Subtract $3 x^{2}$ from $- x^{2}$ .

$- 4 x^{2} + 6 x + 4 x$

Step 1.3.3

Add $6 x$ and $4 x$ .

$- 4 x^{2} + 10 x$

Step 2

Evaluate $f - g$ at $x$ .

$- 4 {(x)}^{2} + 10 (x)$

Step 3

Simplify $- 4 {(x)}^{2} + 10 (x)$ .

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Step 3.1

Remove parentheses.

$- 4 {(x)}^{2} + 10 (x)$

Step 3.2

Multiply $10$ by $x$ .

$- 4 x^{2} + 10 x$