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

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Multiply both sides of the equation by $- \frac{8}{9}$ .

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

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{8}{9} (- \frac{9 y}{8})$ .

Cancel the common factor of $8$ .

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

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

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

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

Factor $8$ out of $- 8$ .

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

Cancel the common factor.

$\frac{8 \cdot - 1}{9} \cdot \frac{- 9 y}{8} = - \frac{8}{9} x$

Rewrite the expression.

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

Cancel the common factor of $9$ .

Factor $9$ out of $- 9 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{8}{9} x$

Multiply $y$ by $1$ .

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

Simplify the right side.

Simplify $- \frac{8}{9} x$ .

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

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

Move $8$ to the left of $x$ .

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

Step 4

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

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

Step 5

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

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

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

Simplify the numerator.

Multiply $- 1$ by $8$ .

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

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

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

Multiply $- 8$ by $9$ .

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

Reduce the expression by cancelling the common factors.

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

Factor $8$ out of $- 72 x$ .

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

Factor $8$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 9 x$ by $1$ .

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

Cancel the common factor of $- 9$ and $9$ .

Factor $9$ out of $- 9 x$ .

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

Cancel the common factors.

Factor $9$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Simplify the numerator.

Multiply $- 1$ by $9$ .

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

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

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

Multiply $- 9$ by $8$ .

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

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 72 x}{9}$ by cancelling the common factors.

Factor $9$ out of $- 72 x$ .

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

Factor $9$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 8 x$ by $1$ .

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

Cancel the common factor of $- 8$ and $8$ .

Factor $8$ out of $- 8 x$ .

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

Cancel the common factors.

Factor $8$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

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

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

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

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