Algebra Examples

$f (x) = - \frac{2 x}{5}$

Step 1

Write $f (x) = - \frac{2 x}{5}$ as an equation.

$y = - \frac{2 x}{5}$

Step 2

Interchange the variables.

$x = - \frac{2 y}{5}$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $- \frac{2 y}{5} = x$ .

$- \frac{2 y}{5} = x$

Step 3.2

Multiply both sides of the equation by $- \frac{5}{2}$ .

$- \frac{5}{2} (- \frac{2 y}{5}) = - \frac{5}{2} x$

Step 3.3

Simplify both sides of the equation.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

Simplify $- \frac{5}{2} (- \frac{2 y}{5})$ .

Tap for more steps...

Step 3.3.1.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.1.1.1.1

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$\frac{- 5}{2} (- \frac{2 y}{5}) = - \frac{5}{2} x$

Step 3.3.1.1.1.2

Move the leading negative in $- \frac{2 y}{5}$ into the numerator.

$\frac{- 5}{2} \cdot \frac{- 2 y}{5} = - \frac{5}{2} x$

Step 3.3.1.1.1.3

Factor $5$ out of $- 5$ .

$\frac{5 (- 1)}{2} \cdot \frac{- 2 y}{5} = - \frac{5}{2} x$

Step 3.3.1.1.1.4

Cancel the common factor.

$\frac{5 \cdot - 1}{2} \cdot \frac{- 2 y}{5} = - \frac{5}{2} x$

Step 3.3.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 y) = - \frac{5}{2} x$

Step 3.3.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.1.2.1

Factor $2$ out of $- 2 y$ .

$\frac{- 1}{2} (2 (- y)) = - \frac{5}{2} x$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- y)) = - \frac{5}{2} x$

Step 3.3.1.1.2.3

Rewrite the expression.

$- - y = - \frac{5}{2} x$

Step 3.3.1.1.3

Multiply.

Tap for more steps...

Step 3.3.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{5}{2} x$

Step 3.3.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{5}{2} x$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Simplify $- \frac{5}{2} x$ .

Tap for more steps...

Step 3.3.2.1.1

Combine $x$ and $\frac{5}{2}$ .

$y = - \frac{x \cdot 5}{2}$

Step 3.3.2.1.2

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

$y = - \frac{5 x}{2}$

Step 4

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

$f^{- 1} (x) = - \frac{5 x}{2}$

Step 5

Verify if $f^{- 1} (x) = - \frac{5 x}{2}$ is the inverse of $f (x) = - \frac{2 x}{5}$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (- \frac{2 x}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{5 (- \frac{2 x}{5})}{2}$

Step 5.2.3

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1

Multiply $- 1$ by $5$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{- 5 \frac{2 x}{5}}{2}$

Step 5.2.3.2

Combine $- 5$ and $\frac{2 x}{5}$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{\frac{- 5 (2 x)}{5}}{2}$

Step 5.2.4

Multiply $- 5$ by $2$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{\frac{- 10 x}{5}}{2}$

Step 5.2.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1

Reduce the expression $\frac{- 10 x}{5}$ by cancelling the common factors.

Tap for more steps...

Step 5.2.5.1.1

Factor $5$ out of $- 10 x$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{\frac{5 (- 2 x)}{5}}{2}$

Step 5.2.5.1.2

Factor $5$ out of $5$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{\frac{5 (- 2 x)}{5 (1)}}{2}$

Step 5.2.5.1.3

Cancel the common factor.

$f^{- 1} (- \frac{2 x}{5}) = - \frac{\frac{5 (- 2 x)}{5 \cdot 1}}{2}$

Step 5.2.5.1.4

Rewrite the expression.

$f^{- 1} (- \frac{2 x}{5}) = - \frac{\frac{- 2 x}{1}}{2}$

Step 5.2.5.2

Divide $- 2 x$ by $1$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{- 2 x}{2}$

Step 5.2.6

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 5.2.6.1

Factor $2$ out of $- 2 x$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{2 (- x)}{2}$

Step 5.2.6.2

Cancel the common factors.

Tap for more steps...

Step 5.2.6.2.1

Factor $2$ out of $2$ .

$f^{- 1} (- \frac{2 x}{5}) = - \frac{2 (- x)}{2 (1)}$

Step 5.2.6.2.2

Cancel the common factor.

$f^{- 1} (- \frac{2 x}{5}) = - \frac{2 (- x)}{2 \cdot 1}$

Step 5.2.6.2.3

Rewrite the expression.

$f^{- 1} (- \frac{2 x}{5}) = - \frac{- x}{1}$

Step 5.2.6.2.4

Divide $- x$ by $1$ .

$f^{- 1} (- \frac{2 x}{5}) = x$

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (- \frac{5 x}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{5 x}{2}) = - \frac{2 (- \frac{5 x}{2})}{5}$

Step 5.3.3

Simplify the numerator.

Tap for more steps...

Step 5.3.3.1

Multiply $- 1$ by $2$ .

$f (- \frac{5 x}{2}) = - \frac{- 2 \frac{5 x}{2}}{5}$

Step 5.3.3.2

Combine $- 2$ and $\frac{5 x}{2}$ .

$f (- \frac{5 x}{2}) = - \frac{\frac{- 2 (5 x)}{2}}{5}$

Step 5.3.4

Multiply $- 2$ by $5$ .

$f (- \frac{5 x}{2}) = - \frac{\frac{- 10 x}{2}}{5}$

Step 5.3.5

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.3.5.1

Reduce the expression $\frac{- 10 x}{2}$ by cancelling the common factors.

Tap for more steps...

Step 5.3.5.1.1

Factor $2$ out of $- 10 x$ .

$f (- \frac{5 x}{2}) = - \frac{\frac{2 (- 5 x)}{2}}{5}$

Step 5.3.5.1.2

Factor $2$ out of $2$ .

$f (- \frac{5 x}{2}) = - \frac{\frac{2 (- 5 x)}{2 (1)}}{5}$

Step 5.3.5.1.3

Cancel the common factor.

$f (- \frac{5 x}{2}) = - \frac{\frac{2 (- 5 x)}{2 \cdot 1}}{5}$

Step 5.3.5.1.4

Rewrite the expression.

$f (- \frac{5 x}{2}) = - \frac{\frac{- 5 x}{1}}{5}$

Step 5.3.5.2

Divide $- 5 x$ by $1$ .

$f (- \frac{5 x}{2}) = - \frac{- 5 x}{5}$

Step 5.3.6

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

Tap for more steps...

Step 5.3.6.1

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

$f (- \frac{5 x}{2}) = - \frac{5 (- x)}{5}$

Step 5.3.6.2

Cancel the common factors.

Tap for more steps...

Step 5.3.6.2.1

Factor $5$ out of $5$ .

$f (- \frac{5 x}{2}) = - \frac{5 (- x)}{5 (1)}$

Step 5.3.6.2.2

Cancel the common factor.

$f (- \frac{5 x}{2}) = - \frac{5 (- x)}{5 \cdot 1}$

Step 5.3.6.2.3

Rewrite the expression.

$f (- \frac{5 x}{2}) = - \frac{- x}{1}$

Step 5.3.6.2.4

Divide $- x$ by $1$ .

$f (- \frac{5 x}{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{5 x}{2}$ is the inverse of $f (x) = - \frac{2 x}{5}$ .

$f^{- 1} (x) = - \frac{5 x}{2}$