Find the x and y Intercepts f(x)=tan(3x)

$f (x) = \tan (3 x)$

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \tan (3 x)$

Solve the equation.

Rewrite the equation as $\tan (3 x) = 0$ .

$\tan (3 x) = 0$

Take the inverse tangent of both sides of the equation to extract $x$ from inside the tangent.

$3 x = \arctan (0)$

Simplify the right side.

The exact value of $\arctan (0)$ is $0$ .

$3 x = 0$

Divide each term in $3 x = 0$ by $3$ and simplify.

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x = 0$

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$3 x = π + 0$

Solve for $x$ .

Add $π$ and $0$ .

$3 x = π$

Divide each term in $3 x = π$ by $3$ and simplify.

Divide each term in $3 x = π$ by $3$ .

$\frac{3 x}{3} = \frac{π}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{π}{3}$

Divide $x$ by $1$ .

$x = \frac{π}{3}$

Find the period of $\tan (3 x)$ .

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

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

$\frac{π}{3}$

The period of the $\tan (3 x)$ function is $\frac{π}{3}$ so values will repeat every $\frac{π}{3}$ radians in both directions.

$x = \frac{π n}{3}, \frac{π}{3} + \frac{π n}{3}$ , for any integer $n$

Consolidate the answers.

$x = \frac{π n}{3}$ , for any integer $n$

x-intercept(s) in point form.

x-intercept(s): $(\frac{π n}{3}, 0)$ , for any integer $n$

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \tan (3 (0))$

Solve the equation.

Remove parentheses.

$y = \tan (3 (0))$

Simplify $\tan (3 (0))$ .

Multiply $3$ by $0$ .

$y = \tan (0)$

The exact value of $\tan (0)$ is $0$ .

$y = 0$

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

List the intersections.

x-intercept(s): $(\frac{π n}{3}, 0)$ , for any integer $n$

y-intercept(s): $(0, 0)$

Find the x and y Intercepts f(x)=tan(3x)

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