Find the Domain of the Composition f(g(x)) f(x)=2/(x-7) , g(x) = square root of x-2

$f (x) = \frac{2}{x - 7}$ , $g (x) = \sqrt{x - 2}$

Set up the composite result function.

$f (g (x))$

Evaluate $f (\sqrt{x - 2})$ by substituting in the value of $g$ into $f$ .

$f (\sqrt{x - 2}) = \frac{2}{(\sqrt{x - 2}) - 7}$

Set the radicand in $\sqrt{x - 2}$ greater than or equal to $0$ to find where the expression is defined.

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

Set the denominator in $\frac{2}{(\sqrt{x - 2}) - 7}$ equal to $0$ to find where the expression is undefined.

$(\sqrt{x - 2}) - 7 = 0$

Solve for $x$ .

Add $7$ to both sides of the equation.

$\sqrt{x - 2} = 7$

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

${\sqrt{x - 2}}^{2} = 7^{2}$

Simplify each side of the equation.

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

${({(x - 2)}^{\frac{1}{2}})}^{2} = 7^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 2)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Rewrite the expression.

${(x - 2)}^{1} = 7^{2}$

Simplify.

$x - 2 = 7^{2}$

Simplify the right side.

Raise $7$ to the power of $2$ .

$x - 2 = 49$

Move all terms not containing $x$ to the right side of the equation.

Add $2$ to both sides of the equation.

$x = 49 + 2$

Add $49$ and $2$ .

$x = 51$

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

Interval Notation:

$[2, 51) \cup (51, \infty)$

Set-Builder Notation:

${x | x \geq 2, x \neq 51}$