Algebra Examples

$f (x) = \frac{3}{5} x + 6$

Step 1

Write $f (x) = \frac{3}{5} x + 6$ as an equation.

$y = \frac{3}{5} x + 6$

Step 2

Interchange the variables.

$x = \frac{3}{5} y + 6$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $\frac{3}{5} y + 6 = x$ .

$\frac{3}{5} y + 6 = x$

Step 3.2

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

$\frac{3 y}{5} + 6 = x$

Step 3.3

Subtract $6$ from both sides of the equation.

$\frac{3 y}{5} = x - 6$

Step 3.4

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

$\frac{5}{3} \cdot \frac{3 y}{5} = \frac{5}{3} (x - 6)$

Step 3.5

Simplify both sides of the equation.

Tap for more steps...

Step 3.5.1

Simplify the left side.

Tap for more steps...

Step 3.5.1.1

Simplify $\frac{5}{3} \cdot \frac{3 y}{5}$ .

Tap for more steps...

Step 3.5.1.1.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.5.1.1.1.1

Cancel the common factor.

$\frac{5}{3} \cdot \frac{3 y}{5} = \frac{5}{3} (x - 6)$

Step 3.5.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{5}{3} (x - 6)$

Step 3.5.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.1.1.2.1

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{5}{3} (x - 6)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{5}{3} (x - 6)$

Step 3.5.1.1.2.3

Rewrite the expression.

$y = \frac{5}{3} (x - 6)$

Step 3.5.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.1

Simplify $\frac{5}{3} (x - 6)$ .

Tap for more steps...

Step 3.5.2.1.1

Apply the distributive property.

$y = \frac{5}{3} x + \frac{5}{3} \cdot - 6$

Step 3.5.2.1.2

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

$y = \frac{5 x}{3} + \frac{5}{3} \cdot - 6$

Step 3.5.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.2.1.3.1

Factor $3$ out of $- 6$ .

$y = \frac{5 x}{3} + \frac{5}{3} \cdot (3 (- 2))$

Step 3.5.2.1.3.2

Cancel the common factor.

$y = \frac{5 x}{3} + \frac{5}{3} \cdot (3 \cdot - 2)$

Step 3.5.2.1.3.3

Rewrite the expression.

$y = \frac{5 x}{3} + 5 \cdot - 2$

Step 3.5.2.1.4

Multiply $5$ by $- 2$ .

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

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 $f (x) = \frac{3}{5} x + 6$ .

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 $f^{- 1} (\frac{3}{5} x + 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{5 (\frac{3}{5} x + 6)}{3} - 10$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1.1

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

$f^{- 1} (\frac{3}{5} x + 6) = \frac{5 (\frac{3 x}{5} + 6)}{3} - 10$

Step 5.2.3.1.2

Factor $3$ out of $\frac{3 x}{5} + 6$ .

Tap for more steps...

Step 5.2.3.1.2.1

Factor $3$ out of $\frac{3 x}{5}$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{5 (3 (\frac{x}{5}) + 6)}{3} - 10$

Step 5.2.3.1.2.2

Factor $3$ out of $6$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{5 (3 (\frac{x}{5}) + 3 \cdot 2)}{3} - 10$

Step 5.2.3.1.2.3

Factor $3$ out of $3 \frac{x}{5} + 3 \cdot 2$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{5 (3 (\frac{x}{5} + 2))}{3} - 10$

$f^{- 1} (\frac{3}{5} x + 6) = \frac{5 \cdot (3 (\frac{x}{5} + 2))}{3} - 10$

Step 5.2.3.1.3

Multiply $5$ by $3$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{15 (\frac{x}{5} + 2)}{3} - 10$

Step 5.2.3.1.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{15 (\frac{x}{5} + 2 \cdot \frac{5}{5})}{3} - 10$

Step 5.2.3.1.5

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

$f^{- 1} (\frac{3}{5} x + 6) = \frac{15 (\frac{x}{5} + \frac{2 \cdot 5}{5})}{3} - 10$

Step 5.2.3.1.6

Combine the numerators over the common denominator.

$f^{- 1} (\frac{3}{5} x + 6) = \frac{15 (\frac{x + 2 \cdot 5}{5})}{3} - 10$

Step 5.2.3.1.7

Multiply $2$ by $5$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{15 (\frac{x + 10}{5})}{3} - 10$

Step 5.2.3.2

Combine $15$ and $\frac{x + 10}{5}$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{\frac{15 (x + 10)}{5}}{3} - 10$

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.2.3.3.1

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

Tap for more steps...

Step 5.2.3.3.1.1

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

$f^{- 1} (\frac{3}{5} x + 6) = \frac{\frac{5 (3 (x + 10))}{5}}{3} - 10$

Step 5.2.3.3.1.2

Factor $5$ out of $5$ .

$f^{- 1} (\frac{3}{5} x + 6) = \frac{\frac{5 (3 (x + 10))}{5 (1)}}{3} - 10$

Step 5.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (\frac{3}{5} x + 6) = \frac{\frac{5 (3 (x + 10))}{5 \cdot 1}}{3} - 10$

Step 5.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (\frac{3}{5} x + 6) = \frac{\frac{3 (x + 10)}{1}}{3} - 10$

Step 5.2.3.3.2

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

$f^{- 1} (\frac{3}{5} x + 6) = \frac{3 (x + 10)}{3} - 10$

Step 5.2.3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (\frac{3}{5} x + 6) = \frac{3 (x + 10)}{3} - 10$

Step 5.2.3.4.2

Divide $x + 10$ by $1$ .

$f^{- 1} (\frac{3}{5} x + 6) = x + 10 - 10$

Step 5.2.4

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

Tap for more steps...

Step 5.2.4.1

Subtract $10$ from $10$ .

$f^{- 1} (\frac{3}{5} x + 6) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\frac{3}{5} x + 6) = x$

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

$f (\frac{5 x}{3} - 10) = \frac{3}{5} \cdot (\frac{5 x}{3} - 10) + 6$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Apply the distributive property.

$f (\frac{5 x}{3} - 10) = \frac{3}{5} \cdot \frac{5 x}{3} + \frac{3}{5} \cdot - 10 + 6$

Step 5.3.3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{5 x}{3} - 10) = \frac{3}{5} \cdot \frac{5 x}{3} + \frac{3}{5} \cdot - 10 + 6$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{5 x}{3} - 10) = \frac{1}{5} \cdot (5 x) + \frac{3}{5} \cdot - 10 + 6$

Step 5.3.3.3

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.3.3.1

Factor $5$ out of $5 x$ .

$f (\frac{5 x}{3} - 10) = \frac{1}{5} \cdot (5 (x)) + \frac{3}{5} \cdot - 10 + 6$

Step 5.3.3.3.2

Cancel the common factor.

$f (\frac{5 x}{3} - 10) = \frac{1}{5} \cdot (5 x) + \frac{3}{5} \cdot - 10 + 6$

Step 5.3.3.3.3

Rewrite the expression.

$f (\frac{5 x}{3} - 10) = x + \frac{3}{5} \cdot - 10 + 6$

Step 5.3.3.4

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.3.3.4.1

Factor $5$ out of $- 10$ .

$f (\frac{5 x}{3} - 10) = x + \frac{3}{5} \cdot (5 (- 2)) + 6$

Step 5.3.3.4.2

Cancel the common factor.

$f (\frac{5 x}{3} - 10) = x + \frac{3}{5} \cdot (5 \cdot - 2) + 6$

Step 5.3.3.4.3

Rewrite the expression.

$f (\frac{5 x}{3} - 10) = x + 3 \cdot - 2 + 6$

Step 5.3.3.5

Multiply $3$ by $- 2$ .

$f (\frac{5 x}{3} - 10) = x - 6 + 6$

Step 5.3.4

Combine the opposite terms in $x - 6 + 6$ .

Tap for more steps...

Step 5.3.4.1

Add $- 6$ and $6$ .

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

Step 5.3.4.2

Add $x$ and $0$ .

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

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 $f (x) = \frac{3}{5} x + 6$ .

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