Find the Intersection of the Functions f(x) = square root of x-3 , G(x) = square root of x+2

$f (x) = \sqrt{x - 3}$ , $G (x) = \sqrt{x + 2}$

Substitute $\sqrt{x + 2}$ for $f (x)$ .

$\sqrt{x + 2} = \sqrt{x - 3}$

Solve for $x$ .

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

${\sqrt{x + 2}}^{2} = {\sqrt{x - 3}}^{2}$

Simplify each side of the equation.

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

${({(x + 2)}^{\frac{1}{2}})}^{2} = {\sqrt{x - 3}}^{2}$

Simplify the left side.

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

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

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

${(x + 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 3}}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 2)}^{\frac{1}{2} \cdot 2} = {\sqrt{x - 3}}^{2}$

Rewrite the expression.

${(x + 2)}^{1} = {\sqrt{x - 3}}^{2}$

Simplify.

$x + 2 = {\sqrt{x - 3}}^{2}$

Simplify the right side.

Rewrite ${\sqrt{x - 3}}^{2}$ as $x - 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x - 3}$ as ${(x - 3)}^{\frac{1}{2}}$ .

$x + 2 = {({(x - 3)}^{\frac{1}{2}})}^{2}$

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

$x + 2 = {(x - 3)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$x + 2 = {(x - 3)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x + 2 = {(x - 3)}^{\frac{2}{2}}$

Rewrite the expression.

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

Simplify.

$x + 2 = x - 3$

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$x + 2 - x = - 3$

Combine the opposite terms in $x + 2 - x$ .

Subtract $x$ from $x$ .

$0 + 2 = - 3$

Add $0$ and $2$ .

$2 = - 3$

Since $2 \neq - 3$ , there are no solutions.

No solution

Find the Intersection of the Functions f(x) = square root of x-3 , G(x) = square root of x+2

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