Graph f(x)=(sin(x))/(|x|)

$f (x) = \frac{\sin (x)}{| x |}$

Find the domain for $f (x) = \frac{\sin (x)}{| x |}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

Set the denominator in $\frac{\sin (x)}{| x |}$ equal to $0$ to find where the expression is undefined.

$| x | = 0$

Solve for $x$ .

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

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.

Substitute the $x$ value $- 2$ into $f (x) = \frac{\sin (x)}{| x |}$ . In this case, the point is $(- 2, - 0.01744974)$ .

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

$f (- 2) = \frac{\sin (- 2)}{| - 2 |}$

Simplify the result.

Evaluate $\sin (- 2)$ .

$f (- 2) = \frac{- 0.03489949}{| - 2 |}$

The absolute value is the distance between a number and zero. The distance between $- 2$ and $0$ is $2$ .

$f (- 2) = \frac{- 0.03489949}{2}$

Divide $- 0.03489949$ by $2$ .

$f (- 2) = - 0.01744974$

The final answer is $- 0.01744974$ .

$y = - 0.01744974$

Substitute the $x$ value $- 1$ into $f (x) = \frac{\sin (x)}{| x |}$ . In this case, the point is $(- 1, - 0.0174524)$ .

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

$f (- 1) = \frac{\sin (- 1)}{| - 1 |}$

Simplify the result.

Evaluate $\sin (- 1)$ .

$f (- 1) = \frac{- 0.0174524}{| - 1 |}$

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

$f (- 1) = \frac{- 0.0174524}{1}$

Divide $- 0.0174524$ by $1$ .

$f (- 1) = - 0.0174524$

The final answer is $- 0.0174524$ .

$y = - 0.0174524$

Substitute the $x$ value $1$ into $f (x) = \frac{\sin (x)}{| x |}$ . In this case, the point is $(1, 0.0174524)$ .

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

$f (1) = \frac{\sin (1)}{| 1 |}$

Simplify the result.

Evaluate $\sin (1)$ .

$f (1) = \frac{0.0174524}{| 1 |}$

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

$f (1) = \frac{0.0174524}{1}$

Divide $0.0174524$ by $1$ .

$f (1) = 0.0174524$

The final answer is $0.0174524$ .

$y = 0.0174524$

Substitute the $x$ value $2$ into $f (x) = \frac{\sin (x)}{| x |}$ . In this case, the point is $(2, 0.01744974)$ .

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

$f (2) = \frac{\sin (2)}{| 2 |}$

Simplify the result.

Evaluate $\sin (2)$ .

$f (2) = \frac{0.03489949}{| 2 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$f (2) = \frac{0.03489949}{2}$

Divide $0.03489949$ by $2$ .

$f (2) = 0.01744974$

The final answer is $0.01744974$ .

$y = 0.01744974$

The absolute value can be graphed using the points around the vertex $(- 2, - 0.02), (- 1, - 0.02), (1, 0.02), (2, 0.02)$

$\begin{array}{c} x & y \\ - 2 & - 0.02 \\ - 1 & - 0.02 \\ 1 & 0.02 \\ 2 & 0.02 \end{array}$

Graph f(x)=(sin(x))/(|x|)

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