Find the Domain of the Composition f(g(x)) f(x)=6/(1-7x) , g(x)=1/x

$f (x) = \frac{6}{1 - 7 x}$ , $g (x) = \frac{1}{x}$

Set up the composite result function.

$f (g (x))$

Evaluate $f (\frac{1}{x})$ by substituting in the value of $g$ into $f$ .

$f (\frac{1}{x}) = \frac{6}{1 - 7 \frac{1}{x}}$

Set the denominator in $\frac{1}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Set the denominator in $\frac{6}{1 - 7 \frac{1}{x}}$ equal to $0$ to find where the expression is undefined.

$1 - 7 \frac{1}{x} = 0$

Solve for $x$ .

Simplify each term.

Combine $- 7$ and $\frac{1}{x}$ .

$1 + \frac{- 7}{x} = 0$

Move the negative in front of the fraction.

$1 - \frac{7}{x} = 0$

Subtract $1$ from both sides of the equation.

$- \frac{7}{x} = - 1$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $- \frac{7}{x} = - 1$ by $x$ to eliminate the fractions.

Multiply each term in $- \frac{7}{x} = - 1$ by $x$ .

$- \frac{7}{x} x = - x$

Simplify the left side.

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{7}{x}$ into the numerator.

$\frac{- 7}{x} x = - x$

Cancel the common factor.

$\frac{- 7}{x} x = - x$

Rewrite the expression.

$- 7 = - x$

Solve the equation.

Rewrite the equation as $- x = - 7$ .

$- x = - 7$

Divide each term in $- x = - 7$ by $- 1$ and simplify.

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x = 7$

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

Interval Notation:

$(- \infty, 0) \cup (0, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq 0, 7}$

Find the Domain of the Composition f(g(x)) f(x)=6/(1-7x) , g(x)=1/x

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