Find the Inverse f(x)=-4/(5x)

$f (x) = - \frac{4}{5 x}$

Step 1

Write $f (x) = - \frac{4}{5 x}$ as an equation.

$y = - \frac{4}{5 x}$

Step 2

Interchange the variables.

$x = - \frac{4}{5 y}$

Step 3

Solve for $y$ .

Rewrite the equation as $- \frac{4}{5 y} = x$ .

$- \frac{4}{5 y} = x$

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.

$5 y, 1$

The LCM of one and any expression is the expression.

$5 y$

Multiply each term in $- \frac{4}{5 y} = x$ by $5 y$ to eliminate the fractions.

Multiply each term in $- \frac{4}{5 y} = x$ by $5 y$ .

$- \frac{4}{5 y} (5 y) = x (5 y)$

Simplify the left side.

Cancel the common factor of $5 y$ .

Move the leading negative in $- \frac{4}{5 y}$ into the numerator.

$\frac{- 4}{5 y} (5 y) = x (5 y)$

Cancel the common factor.

$\frac{- 4}{5 y} (5 y) = x (5 y)$

Rewrite the expression.

$- 4 = x (5 y)$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$- 4 = 5 x y$

Solve the equation.

Rewrite the equation as $5 x y = - 4$ .

$5 x y = - 4$

Divide each term in $5 x y = - 4$ by $5 x$ and simplify.

Divide each term in $5 x y = - 4$ by $5 x$ .

$\frac{5 x y}{5 x} = \frac{- 4}{5 x}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x y}{5 x} = \frac{- 4}{5 x}$

Rewrite the expression.

$\frac{x y}{x} = \frac{- 4}{5 x}$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x y}{x} = \frac{- 4}{5 x}$

Divide $y$ by $1$ .

$y = \frac{- 4}{5 x}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{4}{5 x}$

Step 4

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

$f^{- 1} (x) = - \frac{4}{5 x}$

Step 5

Verify if $f^{- 1} (x) = - \frac{4}{5 x}$ is the inverse of $f (x) = - \frac{4}{5 x}$ .

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 $f^{- 1} (- \frac{4}{5 x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{5 (- \frac{4}{5 x})}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{- 5 \frac{4}{5 x}}$

Combine $- 5$ and $\frac{4}{5 x}$ .

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{\frac{- 5 \cdot 4}{5 x}}$

Multiply $- 5$ by $4$ .

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{\frac{- 20}{5 x}}$

Simplify the denominator.

Reduce the expression $\frac{- 20}{5 x}$ by cancelling the common factors.

Factor $5$ out of $- 20$ .

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{\frac{5 \cdot - 4}{5 x}}$

Factor $5$ out of $5 x$ .

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{\frac{5 \cdot - 4}{5 (x)}}$

Cancel the common factor.

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{\frac{5 \cdot - 4}{5 x}}$

Rewrite the expression.

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{\frac{- 4}{x}}$

Move the negative in front of the fraction.

$f^{- 1} (- \frac{4}{5 x}) = - \frac{4}{- \frac{4}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{4}{5 x}) = - (4 (- \frac{x}{4}))$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$f^{- 1} (- \frac{4}{5 x}) = - (4 (\frac{- x}{4}))$

Cancel the common factor.

$f^{- 1} (- \frac{4}{5 x}) = - (4 (\frac{- x}{4}))$

Rewrite the expression.

$f^{- 1} (- \frac{4}{5 x}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{4}{5 x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{4}{5 x}) = - \frac{4}{5 (- \frac{4}{5 x})}$

Simplify the denominator.

Multiply $- 1$ by $5$ .

$f (- \frac{4}{5 x}) = - \frac{4}{- 5 \frac{4}{5 x}}$

Combine $- 5$ and $\frac{4}{5 x}$ .

$f (- \frac{4}{5 x}) = - \frac{4}{\frac{- 5 \cdot 4}{5 x}}$

Multiply $- 5$ by $4$ .

$f (- \frac{4}{5 x}) = - \frac{4}{\frac{- 20}{5 x}}$

Simplify the denominator.

Reduce the expression $\frac{- 20}{5 x}$ by cancelling the common factors.

Factor $5$ out of $- 20$ .

$f (- \frac{4}{5 x}) = - \frac{4}{\frac{5 \cdot - 4}{5 x}}$

Factor $5$ out of $5 x$ .

$f (- \frac{4}{5 x}) = - \frac{4}{\frac{5 \cdot - 4}{5 (x)}}$

Cancel the common factor.

$f (- \frac{4}{5 x}) = - \frac{4}{\frac{5 \cdot - 4}{5 x}}$

Rewrite the expression.

$f (- \frac{4}{5 x}) = - \frac{4}{\frac{- 4}{x}}$

Move the negative in front of the fraction.

$f (- \frac{4}{5 x}) = - \frac{4}{- \frac{4}{x}}$

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{4}{5 x}) = - (4 (- \frac{x}{4}))$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$f (- \frac{4}{5 x}) = - (4 (\frac{- x}{4}))$

Cancel the common factor.

$f (- \frac{4}{5 x}) = - (4 (\frac{- x}{4}))$

Rewrite the expression.

$f (- \frac{4}{5 x}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{4}{5 x}$ is the inverse of $f (x) = - \frac{4}{5 x}$ .

$f^{- 1} (x) = - \frac{4}{5 x}$

Find the Inverse f(x)=-4/(5x)

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