Algebra Examples

$f (x) = \frac{3 (x + 10)}{5}$

Step 1

Write $f (x) = \frac{3 (x + 10)}{5}$ as an equation.

$y = \frac{3 (x + 10)}{5}$

Step 2

Interchange the variables.

$x = \frac{3 (y + 10)}{5}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3 (y + 10)}{5} = x$ .

$\frac{3 (y + 10)}{5} = x$

Step 3.2

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

$\frac{5}{3} \cdot \frac{3 (y + 10)}{5} = \frac{5}{3} x$

Step 3.3

Simplify both sides of the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify $\frac{5}{3} \cdot \frac{3 (y + 10)}{5}$ .

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Step 3.3.1.1.1

Cancel the common factor of $5$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{5}{3} \cdot \frac{3 (y + 10)}{5} = \frac{5}{3} x$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 (y + 10)) = \frac{5}{3} x$

Step 3.3.1.1.2

Cancel the common factor of $3$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{1}{3} (3 (y + 10)) = \frac{5}{3} x$

Step 3.3.1.1.2.2

Rewrite the expression.

$y + 10 = \frac{5}{3} x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

$y + 10 = \frac{5 x}{3}$

Step 3.4

Subtract $10$ from both sides of the equation.

$y = \frac{5 x}{3} - 10$

Step 4

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

$f^{- 1} (x) = \frac{5 x}{3} - 10$

Step 5

Verify if $f^{- 1} (x) = \frac{5 x}{3} - 10$ is the inverse of $f (x) = \frac{3 (x + 10)}{5}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\frac{3 (x + 10)}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{5 (\frac{3 (x + 10)}{5})}{3} - 10$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Combine $5$ and $\frac{3 (x + 10)}{5}$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{\frac{5 (3 (x + 10))}{5}}{3} - 10$

Step 5.2.3.2

Multiply $5$ by $3$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{\frac{15 (x + 10)}{5}}{3} - 10$

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.3.1

Reduce the expression $\frac{15 (x + 10)}{5}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $5$ out of $15 (x + 10)$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{\frac{5 (3 (x + 10))}{5}}{3} - 10$

Step 5.2.3.3.1.2

Factor $5$ out of $5$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{\frac{5 (3 (x + 10))}{5 (1)}}{3} - 10$

Step 5.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{\frac{5 (3 (x + 10))}{5 \cdot 1}}{3} - 10$

Step 5.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{\frac{3 (x + 10)}{1}}{3} - 10$

Step 5.2.3.3.2

Divide $3 (x + 10)$ by $1$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{3 (x + 10)}{3} - 10$

Step 5.2.3.4

Cancel the common factor of $3$ .

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Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{3 (x + 10)}{5}) = \frac{3 (x + 10)}{3} - 10$

Step 5.2.3.4.2

Divide $x + 10$ by $1$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = x + 10 - 10$

Step 5.2.4

Combine the opposite terms in $x + 10 - 10$ .

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Step 5.2.4.1

Subtract $10$ from $10$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{3 (x + 10)}{5}) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{5 x}{3} - 10)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{5 x}{3} - 10) = \frac{3 ((\frac{5 x}{3} - 10) + 10)}{5}$

Step 5.3.3

Cancel the common factor of $(\frac{5 x}{3} - 10) + 10$ and $5$ .

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Step 5.3.3.1

Factor $5$ out of $3 ((\frac{5 x}{3} - 10) + 10)$ .

$f (\frac{5 x}{3} - 10) = \frac{5 (3 (\frac{x}{3} - 2 + 2))}{5}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $5$ out of $5$ .

$f (\frac{5 x}{3} - 10) = \frac{5 (3 (\frac{x}{3} - 2 + 2))}{5 (1)}$

Step 5.3.3.2.2

Cancel the common factor.

$f (\frac{5 x}{3} - 10) = \frac{5 (3 (\frac{x}{3} - 2 + 2))}{5 \cdot 1}$

Step 5.3.3.2.3

Rewrite the expression.

$f (\frac{5 x}{3} - 10) = \frac{3 (\frac{x}{3} - 2 + 2)}{1}$

Step 5.3.3.2.4

Divide $3 (\frac{x}{3} - 2 + 2)$ by $1$ .

$f (\frac{5 x}{3} - 10) = 3 (\frac{x}{3} - 2 + 2)$

Step 5.3.4

Combine the opposite terms in $\frac{x}{3} - 2 + 2$ .

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Step 5.3.4.1

Add $- 2$ and $2$ .

$f (\frac{5 x}{3} - 10) = 3 (\frac{x}{3} + 0)$

Step 5.3.4.2

Add $\frac{x}{3}$ and $0$ .

$f (\frac{5 x}{3} - 10) = 3 (\frac{x}{3})$

Step 5.3.5

Cancel the common factor of $3$ .

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Step 5.3.5.1

Cancel the common factor.

$f (\frac{5 x}{3} - 10) = 3 (\frac{x}{3})$

Step 5.3.5.2

Rewrite the expression.

$f (\frac{5 x}{3} - 10) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{5 x}{3} - 10$ is the inverse of $f (x) = \frac{3 (x + 10)}{5}$ .

$f^{- 1} (x) = \frac{5 x}{3} - 10$