Precalculus Examples

$g (t) = - (t - 1) \cdot 2 + 5$

Step 1

Consider the difference quotient formula.

$\frac{f (t + h) - f t}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = t + h$ .

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Step 2.1.1

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

$g (t + h) = - ((t + h) - 1) \cdot 2 + 5$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Apply the distributive property.

$g (t + h) = (- t - h + 1) \cdot 2 + 5$

Step 2.1.2.1.2

Multiply $- 1$ by $- 1$ .

$g (t + h) = (- t - h + 1) \cdot 2 + 5$

Step 2.1.2.1.3

Apply the distributive property.

$g (t + h) = - t \cdot 2 - h \cdot 2 + 1 \cdot 2 + 5$

Step 2.1.2.1.4

Simplify.

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Step 2.1.2.1.4.1

Multiply $2$ by $- 1$ .

$g (t + h) = - 2 t - h \cdot 2 + 1 \cdot 2 + 5$

Step 2.1.2.1.4.2

Multiply $2$ by $- 1$ .

$g (t + h) = - 2 t - 2 h + 1 \cdot 2 + 5$

Step 2.1.2.1.4.3

Multiply $2$ by $1$ .

$g (t + h) = - 2 t - 2 h + 2 + 5$

$g (t + h) = - 2 t - 2 h + 2 + 5$

$g (t + h) = - 2 t - 2 h + 2 + 5$

Step 2.1.2.2

Add $2$ and $5$ .

$g (t + h) = - 2 t - 2 h + 7$

Step 2.1.2.3

The final answer is $- 2 t - 2 h + 7$ .

$- 2 t - 2 h + 7$

$- 2 t - 2 h + 7$

$- 2 t - 2 h + 7$

Step 2.2

Reorder $- 2 t$ and $- 2 h$ .

$- 2 h - 2 t + 7$

Step 2.3

Find the components of the definition.

$g (t + h) = - 2 h - 2 t + 7$

$g (t) = - 2 t + 7$

$g (t + h) = - 2 h - 2 t + 7$

$g (t) = - 2 t + 7$

Step 3

Plug in the components.

$\frac{g (t + h) - g t}{h} = \frac{- 2 h - 2 t + 7 - (- 2 t + 7)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- 2 h - 2 t + 7 - (- 2 t) - 1 \cdot 7}{h}$

Step 4.1.2

Multiply $- 2$ by $- 1$ .

$\frac{- 2 h - 2 t + 7 + 2 t - 1 \cdot 7}{h}$

Step 4.1.3

Multiply $- 1$ by $7$ .

$\frac{- 2 h - 2 t + 7 + 2 t - 7}{h}$

Step 4.1.4

Add $- 2 t$ and $2 t$ .

$\frac{- 2 h + 0 + 7 - 7}{h}$

Step 4.1.5

Add $- 2 h$ and $0$ .

$\frac{- 2 h + 7 - 7}{h}$

Step 4.1.6

Subtract $7$ from $7$ .

$\frac{- 2 h + 0}{h}$

Step 4.1.7

Add $- 2 h$ and $0$ .

$\frac{- 2 h}{h}$

$\frac{- 2 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{- 2 h}{h}$

Step 4.2.2

Divide $- 2$ by $1$ .

$- 2$

$- 2$

$- 2$

Step 5

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