Find the Inverse f(x)=-9/(8x)

$f (x) = - \frac{9}{8 x}$

Step 1

Write $f (x) = - \frac{9}{8 x}$ as an equation.

$y = - \frac{9}{8 x}$

Step 2

Interchange the variables.

$x = - \frac{9}{8 y}$

Step 3

Solve for $y$ .

Rewrite the equation as $- \frac{9}{8 y} = x$ .

$- \frac{9}{8 y} = x$

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.

$8 y, 1$

The LCM of one and any expression is the expression.

$8 y$

Multiply each term in $- \frac{9}{8 y} = x$ by $8 y$ to eliminate the fractions.

Multiply each term in $- \frac{9}{8 y} = x$ by $8 y$ .

$- \frac{9}{8 y} (8 y) = x (8 y)$

Simplify the left side.

Cancel the common factor of $8 y$ .

Move the leading negative in $- \frac{9}{8 y}$ into the numerator.

$\frac{- 9}{8 y} (8 y) = x (8 y)$

Cancel the common factor.

$\frac{- 9}{8 y} (8 y) = x (8 y)$

Rewrite the expression.

$- 9 = x (8 y)$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$- 9 = 8 x y$

Solve the equation.

Rewrite the equation as $8 x y = - 9$ .

$8 x y = - 9$

Divide each term in $8 x y = - 9$ by $8 x$ and simplify.

Divide each term in $8 x y = - 9$ by $8 x$ .

$\frac{8 x y}{8 x} = \frac{- 9}{8 x}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x y}{8 x} = \frac{- 9}{8 x}$

Rewrite the expression.

$\frac{x y}{x} = \frac{- 9}{8 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y}{x} = \frac{- 9}{8 x}$

Divide $y$ by $1$ .

$y = \frac{- 9}{8 x}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{9}{8 x}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = - \frac{9}{8 x}$

Step 5

Verify if $f^{- 1} (x) = - \frac{9}{8 x}$ is the inverse of $f (x) = - \frac{9}{8 x}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (- \frac{9}{8 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{8 (- \frac{9}{8 x})}$

Simplify the denominator.

Multiply $- 1$ by $8$ .

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{- 8 \frac{9}{8 x}}$

Combine $- 8$ and $\frac{9}{8 x}$ .

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{\frac{- 8 \cdot 9}{8 x}}$

Multiply $- 8$ by $9$ .

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{\frac{- 72}{8 x}}$

Simplify the denominator.

Reduce the expression $\frac{- 72}{8 x}$ by cancelling the common factors.

Factor $8$ out of $- 72$ .

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{\frac{8 \cdot - 9}{8 x}}$

Factor $8$ out of $8 x$ .

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{\frac{8 \cdot - 9}{8 (x)}}$

Cancel the common factor.

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{\frac{8 \cdot - 9}{8 x}}$

Rewrite the expression.

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{\frac{- 9}{x}}$

Move the negative in front of the fraction.

$f^{- 1} (- \frac{9}{8 x}) = - \frac{9}{- \frac{9}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{9}{8 x}) = - (9 (- \frac{x}{9}))$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$f^{- 1} (- \frac{9}{8 x}) = - (9 (\frac{- x}{9}))$

Cancel the common factor.

$f^{- 1} (- \frac{9}{8 x}) = - (9 (\frac{- x}{9}))$

Rewrite the expression.

$f^{- 1} (- \frac{9}{8 x}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (- \frac{9}{8 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{9}{8 x}) = - \frac{9}{8 (- \frac{9}{8 x})}$

Simplify the denominator.

Multiply $- 1$ by $8$ .

$f (- \frac{9}{8 x}) = - \frac{9}{- 8 \frac{9}{8 x}}$

Combine $- 8$ and $\frac{9}{8 x}$ .

$f (- \frac{9}{8 x}) = - \frac{9}{\frac{- 8 \cdot 9}{8 x}}$

Multiply $- 8$ by $9$ .

$f (- \frac{9}{8 x}) = - \frac{9}{\frac{- 72}{8 x}}$

Simplify the denominator.

Reduce the expression $\frac{- 72}{8 x}$ by cancelling the common factors.

Factor $8$ out of $- 72$ .

$f (- \frac{9}{8 x}) = - \frac{9}{\frac{8 \cdot - 9}{8 x}}$

Factor $8$ out of $8 x$ .

$f (- \frac{9}{8 x}) = - \frac{9}{\frac{8 \cdot - 9}{8 (x)}}$

Cancel the common factor.

$f (- \frac{9}{8 x}) = - \frac{9}{\frac{8 \cdot - 9}{8 x}}$

Rewrite the expression.

$f (- \frac{9}{8 x}) = - \frac{9}{\frac{- 9}{x}}$

Move the negative in front of the fraction.

$f (- \frac{9}{8 x}) = - \frac{9}{- \frac{9}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{9}{8 x}) = - (9 (- \frac{x}{9}))$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$f (- \frac{9}{8 x}) = - (9 (\frac{- x}{9}))$

Cancel the common factor.

$f (- \frac{9}{8 x}) = - (9 (\frac{- x}{9}))$

Rewrite the expression.

$f (- \frac{9}{8 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{9}{8 x}$ is the inverse of $f (x) = - \frac{9}{8 x}$ .

$f^{- 1} (x) = - \frac{9}{8 x}$

Find the Inverse f(x)=-9/(8x)

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