Calculus Examples

$f (x) = | x^{2} - 1 |$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

For each $x$ value, there is one $y$ value. Select a few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

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Step 2.1

Substitute the $x$ value $- 2$ into $f (x) = | x^{2} - 1 |$ . In this case, the point is $(- 2, 3)$ .

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Step 2.1.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = | {(- 2)}^{2} - 1 |$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

$f (- 2) = | 4 - 1 |$

Step 2.1.2.2

Subtract $1$ from $4$ .

$f (- 2) = | 3 |$

Step 2.1.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$f (- 2) = 3$

Step 2.1.2.4

The final answer is $3$ .

$y = 3$

Step 2.2

Substitute the $x$ value $- 1$ into $f (x) = | x^{2} - 1 |$ . In this case, the point is $(- 1, 0)$ .

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Step 2.2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = | {(- 1)}^{2} - 1 |$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

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

$f (- 1) = | 1 - 1 |$

Step 2.2.2.2

Subtract $1$ from $1$ .

$f (- 1) = | 0 |$

Step 2.2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$f (- 1) = 0$

Step 2.2.2.4

The final answer is $0$ .

$y = 0$

Step 2.3

Substitute the $x$ value $0$ into $f (x) = | x^{2} - 1 |$ . In this case, the point is $(0, 1)$ .

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Step 2.3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = | {(0)}^{2} - 1 |$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = | 0 - 1 |$

Step 2.3.2.2

Subtract $1$ from $0$ .

$f (0) = | - 1 |$

Step 2.3.2.3

The absolute value is the distance between a number and zero. The distance between $- 1$ and $0$ is $1$ .

$f (0) = 1$

Step 2.3.2.4

The final answer is $1$ .

$y = 1$

Step 2.4

Substitute the $x$ value $1$ into $f (x) = | x^{2} - 1 |$ . In this case, the point is $(1, 0)$ .

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Step 2.4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = | {(1)}^{2} - 1 |$

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

One to any power is one.

$f (1) = | 1 - 1 |$

Step 2.4.2.2

Subtract $1$ from $1$ .

$f (1) = | 0 |$

Step 2.4.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$f (1) = 0$

Step 2.4.2.4

The final answer is $0$ .

$y = 0$

Step 2.5

The absolute value can be graphed using the points around the vertex $(- 2, 3), (- 1, 0), (0, 1), (1, 0)$

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{array}$

Step 3