Find the Inverse f(x)=-(3x)/5

$f (x) = - \frac{3 x}{5}$

Step 1

Write $f (x) = - \frac{3 x}{5}$ as an equation.

$y = - \frac{3 x}{5}$

Step 2

Interchange the variables.

$x = - \frac{3 y}{5}$

Step 3

Solve for $y$ .

Rewrite the equation as $- \frac{3 y}{5} = x$ .

$- \frac{3 y}{5} = x$

Multiply both sides of the equation by $- \frac{5}{3}$ .

$- \frac{5}{3} (- \frac{3 y}{5}) = - \frac{5}{3} x$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{5}{3} (- \frac{3 y}{5})$ .

Cancel the common factor of $5$ .

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$\frac{- 5}{3} (- \frac{3 y}{5}) = - \frac{5}{3} x$

Move the leading negative in $- \frac{3 y}{5}$ into the numerator.

$\frac{- 5}{3} \cdot \frac{- 3 y}{5} = - \frac{5}{3} x$

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{3} \cdot \frac{- 3 y}{5} = - \frac{5}{3} x$

Cancel the common factor.

$\frac{5 \cdot - 1}{3} \cdot \frac{- 3 y}{5} = - \frac{5}{3} x$

Rewrite the expression.

$\frac{- 1}{3} (- 3 y) = - \frac{5}{3} x$

Cancel the common factor of $3$ .

Factor $3$ out of $- 3 y$ .

$\frac{- 1}{3} (3 (- y)) = - \frac{5}{3} x$

Cancel the common factor.

$\frac{- 1}{3} (3 (- y)) = - \frac{5}{3} x$

Rewrite the expression.

$- - y = - \frac{5}{3} x$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{5}{3} x$

Multiply $y$ by $1$ .

$y = - \frac{5}{3} x$

Simplify the right side.

Simplify $- \frac{5}{3} x$ .

Combine $x$ and $\frac{5}{3}$ .

$y = - \frac{x \cdot 5}{3}$

Move $5$ to the left of $x$ .

$y = - \frac{5 x}{3}$

Step 4

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

$f^{- 1} (x) = - \frac{5 x}{3}$

Step 5

Verify if $f^{- 1} (x) = - \frac{5 x}{3}$ is the inverse of $f (x) = - \frac{3 x}{5}$ .

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 $f^{- 1} (- \frac{3 x}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{5 (- \frac{3 x}{5})}{3}$

Simplify the numerator.

Multiply $- 1$ by $5$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{- 5 \frac{3 x}{5}}{3}$

Combine $- 5$ and $\frac{3 x}{5}$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{\frac{- 5 (3 x)}{5}}{3}$

Multiply $- 5$ by $3$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{\frac{- 15 x}{5}}{3}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 15 x}{5}$ by cancelling the common factors.

Factor $5$ out of $- 15 x$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{\frac{5 (- 3 x)}{5}}{3}$

Factor $5$ out of $5$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{\frac{5 (- 3 x)}{5 (1)}}{3}$

Cancel the common factor.

$f^{- 1} (- \frac{3 x}{5}) = - \frac{\frac{5 (- 3 x)}{5 \cdot 1}}{3}$

Rewrite the expression.

$f^{- 1} (- \frac{3 x}{5}) = - \frac{\frac{- 3 x}{1}}{3}$

Divide $- 3 x$ by $1$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{- 3 x}{3}$

Cancel the common factor of $- 3$ and $3$ .

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

$f^{- 1} (- \frac{3 x}{5}) = - \frac{3 (- x)}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$f^{- 1} (- \frac{3 x}{5}) = - \frac{3 (- x)}{3 (1)}$

Cancel the common factor.

$f^{- 1} (- \frac{3 x}{5}) = - \frac{3 (- x)}{3 \cdot 1}$

Rewrite the expression.

$f^{- 1} (- \frac{3 x}{5}) = - \frac{- x}{1}$

Divide $- x$ by $1$ .

$f^{- 1} (- \frac{3 x}{5}) = x$

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

Set up the composite result function.

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

Evaluate $f (- \frac{5 x}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{5 x}{3}) = - \frac{3 (- \frac{5 x}{3})}{5}$

Simplify the numerator.

Multiply $- 1$ by $3$ .

$f (- \frac{5 x}{3}) = - \frac{- 3 \frac{5 x}{3}}{5}$

Combine $- 3$ and $\frac{5 x}{3}$ .

$f (- \frac{5 x}{3}) = - \frac{\frac{- 3 (5 x)}{3}}{5}$

Multiply $- 3$ by $5$ .

$f (- \frac{5 x}{3}) = - \frac{\frac{- 15 x}{3}}{5}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{- 15 x}{3}$ by cancelling the common factors.

Factor $3$ out of $- 15 x$ .

$f (- \frac{5 x}{3}) = - \frac{\frac{3 (- 5 x)}{3}}{5}$

Factor $3$ out of $3$ .

$f (- \frac{5 x}{3}) = - \frac{\frac{3 (- 5 x)}{3 (1)}}{5}$

Cancel the common factor.

$f (- \frac{5 x}{3}) = - \frac{\frac{3 (- 5 x)}{3 \cdot 1}}{5}$

Rewrite the expression.

$f (- \frac{5 x}{3}) = - \frac{\frac{- 5 x}{1}}{5}$

Divide $- 5 x$ by $1$ .

$f (- \frac{5 x}{3}) = - \frac{- 5 x}{5}$

Cancel the common factor of $- 5$ and $5$ .

Factor $5$ out of $- 5 x$ .

$f (- \frac{5 x}{3}) = - \frac{5 (- x)}{5}$

Cancel the common factors.

Factor $5$ out of $5$ .

$f (- \frac{5 x}{3}) = - \frac{5 (- x)}{5 (1)}$

Cancel the common factor.

$f (- \frac{5 x}{3}) = - \frac{5 (- x)}{5 \cdot 1}$

Rewrite the expression.

$f (- \frac{5 x}{3}) = - \frac{- x}{1}$

Divide $- x$ by $1$ .

$f (- \frac{5 x}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{5 x}{3}$ is the inverse of $f (x) = - \frac{3 x}{5}$ .

$f^{- 1} (x) = - \frac{5 x}{3}$

Find the Inverse f(x)=-(3x)/5

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