Precalculus Examples

Find $f (x + h)$ for $f (x) = x^{2} + 2 x$

Step 1

Replace the variable $x$ with $x + h$ in the expression.

$f (x + h) = {(x + h)}^{2} + 2 (x + h)$

Step 2

Simplify ${(x + h)}^{2} + 2 (x + h)$ .

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Step 2.1

Remove parentheses.

${(x + h)}^{2} + 2 (x + h)$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$(x + h) (x + h) + 2 (x + h)$

Step 2.2.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$x (x + h) + h (x + h) + 2 (x + h)$

Step 2.2.2.2

Apply the distributive property.

$x \cdot x + x h + h (x + h) + 2 (x + h)$

Step 2.2.2.3

Apply the distributive property.

$x \cdot x + x h + h x + h \cdot h + 2 (x + h)$

$x \cdot x + x h + h x + h \cdot h + 2 (x + h)$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x h + h x + h \cdot h + 2 (x + h)$

Step 2.2.3.1.2

Multiply $h$ by $h$ .

$x^{2} + x h + h x + h^{2} + 2 (x + h)$

$x^{2} + x h + h x + h^{2} + 2 (x + h)$

Step 2.2.3.2

Add $x h$ and $h x$ .

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Step 2.2.3.2.1

Reorder $x$ and $h$ .

$x^{2} + h x + h x + h^{2} + 2 (x + h)$

Step 2.2.3.2.2

Add $h x$ and $h x$ .

$x^{2} + 2 h x + h^{2} + 2 (x + h)$

$x^{2} + 2 h x + h^{2} + 2 (x + h)$

$x^{2} + 2 h x + h^{2} + 2 (x + h)$

Step 2.2.4

Apply the distributive property.

$x^{2} + 2 h x + h^{2} + 2 x + 2 h$

$x^{2} + 2 h x + h^{2} + 2 x + 2 h$

Step 2.3

Simplify the expression.

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Step 2.3.1

Move $2 x$ .

$x^{2} + 2 h x + h^{2} + 2 h + 2 x$

Step 2.3.2

Move $x^{2}$ .

$2 h x + h^{2} + x^{2} + 2 h + 2 x$

Step 2.3.3

Reorder $2 h x$ and $h^{2}$ .

$h^{2} + 2 h x + x^{2} + 2 h + 2 x$

$h^{2} + 2 h x + x^{2} + 2 h + 2 x$

$h^{2} + 2 h x + x^{2} + 2 h + 2 x$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$