Find the x and y Intercepts f(x) = square root of 4-x^2

$f (x) = \sqrt{4 - x^{2}}$

Step 1

Find the x-intercepts.

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

$0 = \sqrt{4 - x^{2}}$

Solve the equation.

Rewrite the equation as $\sqrt{4 - x^{2}} = 0$ .

$\sqrt{4 - x^{2}} = 0$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{4 - x^{2}}}^{2} = 0^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

${({(4 - x^{2})}^{\frac{1}{2}})}^{2} = 0^{2}$

Simplify the left side.

Simplify ${({(4 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(4 - x^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(4 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(4 - x^{2})}^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

${(4 - x^{2})}^{1} = 0^{2}$

Simplify.

$4 - x^{2} = 0^{2}$

Simplify the right side.

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

$4 - x^{2} = 0$

Solve for $x$ .

Subtract $4$ from both sides of the equation.

$- x^{2} = - 4$

Divide each term in $- x^{2} = - 4$ by $- 1$ and simplify.

Divide each term in $- x^{2} = - 4$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 4}{- 1}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x^{2} = 4$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

x-intercept(s) in point form.

x-intercept(s): $(2, 0), (- 2, 0)$

Step 2

Find the y-intercepts.

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

$y = \sqrt{4 - {(0)}^{2}}$

Simplify $\sqrt{4 - {(0)}^{2}}$ .

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

$y = \sqrt{4 - 0}$

Multiply $- 1$ by $0$ .

$y = \sqrt{4 + 0}$

Add $4$ and $0$ .

$y = \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$y = \sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$y = 2$

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(2, 0), (- 2, 0)$

y-intercept(s): $(0, 2)$

Step 4

Find the x and y Intercepts f(x) = square root of 4-x^2

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