Find the Domain of the Composition f(g(x)) f(x)=1/(-10x-7) , g(x)=1/(-6x+5)

$f (x) = \frac{1}{- 10 x - 7}$ , $g (x) = \frac{1}{- 6 x + 5}$

Set up the composite result function.

$f (g (x))$

Evaluate $f (\frac{1}{- 6 x + 5})$ by substituting in the value of $g$ into $f$ .

$f (\frac{1}{- 6 x + 5}) = \frac{1}{- 10 \frac{1}{- 6 x + 5} - 7}$

Set the denominator in $\frac{1}{- 6 x + 5}$ equal to $0$ to find where the expression is undefined.

$- 6 x + 5 = 0$

Solve for $x$ .

Subtract $5$ from both sides of the equation.

$- 6 x = - 5$

Divide each term in $- 6 x = - 5$ by $- 6$ and simplify.

Divide each term in $- 6 x = - 5$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 5}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 5}{- 6}$

Divide $x$ by $1$ .

$x = \frac{- 5}{- 6}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{5}{6}$

Set the denominator in $\frac{1}{- 10 \frac{1}{- 6 x + 5} - 7}$ equal to $0$ to find where the expression is undefined.

$- 10 \frac{1}{- 6 x + 5} - 7 = 0$

Solve for $x$ .

Simplify $- 10 \frac{1}{- 6 x + 5} - 7$ .

Simplify each term.

Combine $- 10$ and $\frac{1}{- 6 x + 5}$ .

$\frac{- 10}{- 6 x + 5} - 7 = 0$

Move the negative in front of the fraction.

$- \frac{10}{- 6 x + 5} - 7 = 0$

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{- 6 x + 5}{- 6 x + 5}$ .

$- \frac{10}{- 6 x + 5} - 7 \cdot \frac{- 6 x + 5}{- 6 x + 5} = 0$

Simplify terms.

Combine $- 7$ and $\frac{- 6 x + 5}{- 6 x + 5}$ .

$- \frac{10}{- 6 x + 5} + \frac{- 7 (- 6 x + 5)}{- 6 x + 5} = 0$

Combine the numerators over the common denominator.

$\frac{- 10 - 7 (- 6 x + 5)}{- 6 x + 5} = 0$

Simplify the numerator.

Factor $- 1$ out of $- 10 - 7 (- 6 x + 5)$ .

Reorder $- 10$ and $- 7 (- 6 x + 5)$ .

$\frac{- 7 (- 6 x + 5) - 10}{- 6 x + 5} = 0$

Factor $- 1$ out of $- 7 (- 6 x + 5)$ .

$\frac{- (7 (- 6 x + 5)) - 10}{- 6 x + 5} = 0$

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{- (7 (- 6 x + 5)) - 1 (10)}{- 6 x + 5} = 0$

Factor $- 1$ out of $- (7 (- 6 x + 5)) - 1 (10)$ .

$\frac{- (7 (- 6 x + 5) + 10)}{- 6 x + 5} = 0$

Apply the distributive property.

$\frac{- (7 (- 6 x) + 7 \cdot 5 + 10)}{- 6 x + 5} = 0$

Multiply $- 6$ by $7$ .

$\frac{- (- 42 x + 7 \cdot 5 + 10)}{- 6 x + 5} = 0$

Multiply $7$ by $5$ .

$\frac{- (- 42 x + 35 + 10)}{- 6 x + 5} = 0$

Add $35$ and $10$ .

$\frac{- (- 42 x + 45)}{- 6 x + 5} = 0$

Factor $3$ out of $- 42 x + 45$ .

Factor $3$ out of $- 42 x$ .

$\frac{- (3 (- 14 x) + 45)}{- 6 x + 5} = 0$

Factor $3$ out of $45$ .

$\frac{- (3 (- 14 x) + 3 (15))}{- 6 x + 5} = 0$

Factor $3$ out of $3 (- 14 x) + 3 (15)$ .

$\frac{- (3 (- 14 x + 15))}{- 6 x + 5} = 0$

$\frac{- 1 \cdot 3 (- 14 x + 15)}{- 6 x + 5} = 0$

Multiply $- 1$ by $3$ .

$\frac{- 3 (- 14 x + 15)}{- 6 x + 5} = 0$

Simplify terms.

Move the negative in front of the fraction.

$- \frac{(3) (- 14 x + 15)}{- 6 x + 5} = 0$

Factor $- 1$ out of $- 14 x$ .

$- \frac{3 (- (14 x) + 15)}{- 6 x + 5} = 0$

Rewrite $15$ as $- 1 (- 15)$ .

$- \frac{3 (- (14 x) - 1 (- 15))}{- 6 x + 5} = 0$

Factor $- 1$ out of $- (14 x) - 1 (- 15)$ .

$- \frac{3 (- (14 x - 15))}{- 6 x + 5} = 0$

Rewrite $- (14 x - 15)$ as $- 1 (14 x - 15)$ .

$- \frac{3 (- 1 (14 x - 15))}{- 6 x + 5} = 0$

Factor $- 1$ out of $- 6 x$ .

$- \frac{3 (- 1 (14 x - 15))}{- (6 x) + 5} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$- \frac{3 (- 1 (14 x - 15))}{- (6 x) - 1 (- 5)} = 0$

Factor $- 1$ out of $- (6 x) - 1 (- 5)$ .

$- \frac{3 (- 1 (14 x - 15))}{- (6 x - 5)} = 0$

Rewrite $- (6 x - 5)$ as $- 1 (6 x - 5)$ .

$- \frac{3 (- 1 (14 x - 15))}{- 1 (6 x - 5)} = 0$

Cancel the common factor.

$- \frac{3 (- 1 (14 x - 15))}{- 1 (6 x - 5)} = 0$

Rewrite the expression.

$- \frac{3 (14 x - 15)}{6 x - 5} = 0$

Set the numerator equal to zero.

$3 (14 x - 15) = 0$

Solve the equation for $x$ .

Divide each term in $3 (14 x - 15) = 0$ by $3$ and simplify.

Divide each term in $3 (14 x - 15) = 0$ by $3$ .

$\frac{3 (14 x - 15)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (14 x - 15)}{3} = \frac{0}{3}$

Divide $14 x - 15$ by $1$ .

$14 x - 15 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$14 x - 15 = 0$

Add $15$ to both sides of the equation.

$14 x = 15$

Divide each term in $14 x = 15$ by $14$ and simplify.

Divide each term in $14 x = 15$ by $14$ .

$\frac{14 x}{14} = \frac{15}{14}$

Simplify the left side.

Cancel the common factor of $14$ .

Cancel the common factor.

$\frac{14 x}{14} = \frac{15}{14}$

Divide $x$ by $1$ .

$x = \frac{15}{14}$

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

Interval Notation:

$(- \infty, \frac{5}{6}) \cup (\frac{5}{6}, \frac{15}{14}) \cup (\frac{15}{14}, \infty)$

Set-Builder Notation:

${x | x \neq \frac{5}{6}, \frac{15}{14}}$

Find the Domain of the Composition f(g(x)) f(x)=1/(-10x-7) , g(x)=1/(-6x+5)

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