Precalculus Examples

If $f (x) = 2 x$ and $g (x) = 2 x^{2} - 1$ find $g (f (- 1))$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (2 x)$ by substituting in the value of $f$ into $g$ .

$g (2 x) = 2 {(2 x)}^{2} - 1$

Step 3

Simplify each term.

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Step 3.1

Apply the product rule to $2 x$ .

$g (2 x) = 2 (2^{2} x^{2}) - 1$

Step 3.2

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 3.2.1

Move $2^{2}$ .

$g (2 x) = 2^{2} \cdot (2 x^{2}) - 1$

Step 3.2.2

Multiply $2^{2}$ by $2$ .

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Step 3.2.2.1

Raise $2$ to the power of $1$ .

$g (2 x) = 2^{2} \cdot (2 x^{2}) - 1$

Step 3.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$g (2 x) = 2^{2 + 1} x^{2} - 1$

Step 3.2.3

Add $2$ and $1$ .

$g (2 x) = 2^{3} x^{2} - 1$

Step 3.3

Raise $2$ to the power of $3$ .

$g (2 x) = 8 x^{2} - 1$

Step 4

Evaluate the result function by replacing $x$ with $- 1$ in the function.

$8 {(- 1)}^{2} - 1$

Step 5

Simplify the evaluated function.

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

$8 \cdot 1 - 1$

Step 5.1.2

Multiply $8$ by $1$ .

$8 - 1$

Step 5.2

Subtract $1$ from $8$ .

$7$