Algebra Examples

$\frac{- 2 x}{17}$

Step 1

Interchange the variables.

$x = \frac{- 2 y}{17}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{- 2 y}{17} = x$ .

$\frac{- 2 y}{17} = x$

Step 2.2

Multiply both sides of the equation by $\frac{17}{- 2}$ .

$\frac{17}{- 2} \cdot \frac{- 2 y}{17} = \frac{17}{- 2} x$

Step 2.3

Simplify both sides of the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{17}{- 2} \cdot \frac{- 2 y}{17}$ .

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Step 2.3.1.1.1

Cancel the common factor of $17$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

$\frac{17}{- 2} \cdot \frac{- 2 y}{17} = \frac{17}{- 2} x$

Step 2.3.1.1.1.2

Rewrite the expression.

$\frac{1}{- 2} (- 2 y) = \frac{17}{- 2} x$

Step 2.3.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.1.1.2.1

Factor $2$ out of $- 2$ .

$\frac{1}{2 (- 1)} (- 2 y) = \frac{17}{- 2} x$

Step 2.3.1.1.2.2

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

$\frac{1}{2 (- 1)} (2 (- y)) = \frac{17}{- 2} x$

Step 2.3.1.1.2.3

Cancel the common factor.

$\frac{1}{2 \cdot - 1} (2 (- y)) = \frac{17}{- 2} x$

Step 2.3.1.1.2.4

Rewrite the expression.

$\frac{1}{- 1} (- y) = \frac{17}{- 2} x$

Step 2.3.1.1.3

Combine $\frac{1}{- 1}$ and $y$ .

$- \frac{y}{- 1} = \frac{17}{- 2} x$

Step 2.3.1.1.4

Simplify the expression.

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Step 2.3.1.1.4.1

Move the negative one from the denominator of $\frac{y}{- 1}$ .

$- (- 1 \cdot y) = \frac{17}{- 2} x$

Step 2.3.1.1.4.2

Rewrite $- 1 \cdot y$ as $- y$ .

$- - y = \frac{17}{- 2} x$

Step 2.3.1.1.5

Multiply $- - y$ .

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Step 2.3.1.1.5.1

Multiply $- 1$ by $- 1$ .

$1 y = \frac{17}{- 2} x$

Step 2.3.1.1.5.2

Multiply $y$ by $1$ .

$y = \frac{17}{- 2} x$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\frac{17}{- 2} x$ .

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Step 2.3.2.1.1

Move the negative in front of the fraction.

$y = - \frac{17}{2} x$

Step 2.3.2.1.2

Combine $x$ and $\frac{17}{2}$ .

$y = - \frac{x \cdot 17}{2}$

Step 2.3.2.1.3

Move $17$ to the left of $x$ .

$y = - \frac{17 x}{2}$

Step 3

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

$f^{- 1} (x) = - \frac{17 x}{2}$

Step 4

Verify if $f^{- 1} (x) = - \frac{17 x}{2}$ is the inverse of $f (x) = \frac{- 2 x}{17}$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\frac{- 2 x}{17})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{17 (\frac{- 2 x}{17})}{2}$

Step 4.2.3

Combine $17$ and $\frac{- 2 x}{17}$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{\frac{17 (- 2 x)}{17}}{2}$

Step 4.2.4

Multiply $- 2$ by $17$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{\frac{- 34 x}{17}}{2}$

Step 4.2.5

Reduce the expression by cancelling the common factors.

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Step 4.2.5.1

Reduce the expression $\frac{- 34 x}{17}$ by cancelling the common factors.

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Step 4.2.5.1.1

Factor $17$ out of $- 34 x$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{\frac{17 (- 2 x)}{17}}{2}$

Step 4.2.5.1.2

Factor $17$ out of $17$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{\frac{17 (- 2 x)}{17 (1)}}{2}$

Step 4.2.5.1.3

Cancel the common factor.

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{\frac{17 (- 2 x)}{17 \cdot 1}}{2}$

Step 4.2.5.1.4

Rewrite the expression.

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{\frac{- 2 x}{1}}{2}$

Step 4.2.5.2

Divide $- 2 x$ by $1$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{- 2 x}{2}$

Step 4.2.6

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

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Step 4.2.6.1

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

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{2 (- x)}{2}$

Step 4.2.6.2

Cancel the common factors.

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Step 4.2.6.2.1

Factor $2$ out of $2$ .

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{2 (- x)}{2 (1)}$

Step 4.2.6.2.2

Cancel the common factor.

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{2 (- x)}{2 \cdot 1}$

Step 4.2.6.2.3

Rewrite the expression.

$f^{- 1} (\frac{- 2 x}{17}) = - \frac{- x}{1}$

Step 4.2.6.2.4

Divide $- x$ by $1$ .

$f^{- 1} (\frac{- 2 x}{17}) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \frac{17 x}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{17 x}{2}) = \frac{- 2 (- \frac{17 x}{2})}{17}$

Step 4.3.3

Simplify the numerator.

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Step 4.3.3.1

Multiply $- 1$ by $- 2$ .

$f (- \frac{17 x}{2}) = \frac{2 (\frac{17 x}{2})}{17}$

Step 4.3.3.2

Combine $2$ and $\frac{17 x}{2}$ .

$f (- \frac{17 x}{2}) = \frac{\frac{2 (17 x)}{2}}{17}$

Step 4.3.4

Multiply $2$ by $17$ .

$f (- \frac{17 x}{2}) = \frac{\frac{34 x}{2}}{17}$

Step 4.3.5

Reduce the expression by cancelling the common factors.

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Step 4.3.5.1

Reduce the expression $\frac{34 x}{2}$ by cancelling the common factors.

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Step 4.3.5.1.1

Factor $2$ out of $34 x$ .

$f (- \frac{17 x}{2}) = \frac{\frac{2 (17 x)}{2}}{17}$

Step 4.3.5.1.2

Factor $2$ out of $2$ .

$f (- \frac{17 x}{2}) = \frac{\frac{2 (17 x)}{2 (1)}}{17}$

Step 4.3.5.1.3

Cancel the common factor.

$f (- \frac{17 x}{2}) = \frac{\frac{2 (17 x)}{2 \cdot 1}}{17}$

Step 4.3.5.1.4

Rewrite the expression.

$f (- \frac{17 x}{2}) = \frac{\frac{17 x}{1}}{17}$

Step 4.3.5.2

Divide $17 x$ by $1$ .

$f (- \frac{17 x}{2}) = \frac{17 x}{17}$

Step 4.3.6

Cancel the common factor of $17$ .

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Step 4.3.6.1

Cancel the common factor.

$f (- \frac{17 x}{2}) = \frac{17 x}{17}$

Step 4.3.6.2

Divide $x$ by $1$ .

$f (- \frac{17 x}{2}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{17 x}{2}$ is the inverse of $f (x) = \frac{- 2 x}{17}$ .

$f^{- 1} (x) = - \frac{17 x}{2}$