Precalculus Examples

Find $(g - h) (x)$ when $g (x) = x^{2} - 7 x + 11$ and $h (x) = x^{3} + 7 x + 1$

Step 1

Evaluate $g - h$ .

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Step 1.1

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

$(x^{2} - 7 x + 11) - (x^{3} + 7 x + 1)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$x^{2} - 7 x + 11 - x^{3} - (7 x) - 1 \cdot 1$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Multiply $7$ by $- 1$ .

$x^{2} - 7 x + 11 - x^{3} - 7 x - 1 \cdot 1$

Step 1.2.2.2

Multiply $- 1$ by $1$ .

$x^{2} - 7 x + 11 - x^{3} - 7 x - 1$

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Subtract $7 x$ from $- 7 x$ .

$x^{2} - 14 x + 11 - x^{3} - 1$

Step 1.3.2

Simplify the expression.

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Step 1.3.2.1

Subtract $1$ from $11$ .

$x^{2} - 14 x - x^{3} + 10$

Step 1.3.2.2

Move $- 14 x$ .

$x^{2} - x^{3} - 14 x + 10$

Step 1.3.2.3

Reorder $x^{2}$ and $- x^{3}$ .

$- x^{3} + x^{2} - 14 x + 10$

Step 2

Evaluate $g - h$ at $x$ .

$- {(x)}^{3} + {(x)}^{2} - 14 x + 10$

Step 3

Simplify $- {(x)}^{3} + {(x)}^{2} - 14 x + 10$ .

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Step 3.1

Remove parentheses.

$- {(x)}^{3} + {(x)}^{2} - 14 x + 10$

Step 3.2

Remove parentheses.

$- x^{3} + x^{2} - 14 x + 10$