Precalculus Examples

$f (x) = - \frac{7}{2} x - 3$

Step 1

Write $f (x) = - \frac{7}{2} x - 3$ as an equation.

$y = - \frac{7}{2} x - 3$

Step 2

Interchange the variables.

$x = - \frac{7}{2} y - 3$

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $- \frac{7}{2} y - 3 = x$ .

$- \frac{7}{2} y - 3 = x$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Combine $y$ and $\frac{7}{2}$ .

$- \frac{y \cdot 7}{2} - 3 = x$

Step 3.2.2

Move $7$ to the left of $y$ .

$- \frac{7 y}{2} - 3 = x$

Step 3.3

Add $3$ to both sides of the equation.

$- \frac{7 y}{2} = x + 3$

Step 3.4

Multiply both sides of the equation by $- \frac{2}{7}$ .

$- \frac{2}{7} (- \frac{7 y}{2}) = - \frac{2}{7} (x + 3)$

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 $- \frac{2}{7} (- \frac{7 y}{2})$ .

Tap for more steps...

Step 3.5.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.1.1.1.1

Move the leading negative in $- \frac{2}{7}$ into the numerator.

$\frac{- 2}{7} (- \frac{7 y}{2}) = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{7 y}{2}$ into the numerator.

$\frac{- 2}{7} \cdot \frac{- 7 y}{2} = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.1.3

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{7} \cdot \frac{- 7 y}{2} = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.1.4

Cancel the common factor.

$\frac{2 \cdot - 1}{7} \cdot \frac{- 7 y}{2} = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{7} (- 7 y) = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.5.1.1.2.1

Factor $7$ out of $- 7 y$ .

$\frac{- 1}{7} (7 (- y)) = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{7} (7 (- y)) = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.2.3

Rewrite the expression.

$- - y = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.3

Multiply.

Tap for more steps...

Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 y = - \frac{2}{7} (x + 3)$

Step 3.5.1.1.3.2

Multiply $y$ by $1$ .

$y = - \frac{2}{7} (x + 3)$

Step 3.5.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.1

Simplify $- \frac{2}{7} (x + 3)$ .

Tap for more steps...

Step 3.5.2.1.1

Apply the distributive property.

$y = - \frac{2}{7} x - \frac{2}{7} \cdot 3$

Step 3.5.2.1.2

Combine $x$ and $\frac{2}{7}$ .

$y = - \frac{x \cdot 2}{7} - \frac{2}{7} \cdot 3$

Step 3.5.2.1.3

Multiply $- \frac{2}{7} \cdot 3$ .

Tap for more steps...

Step 3.5.2.1.3.1

Multiply $3$ by $- 1$ .

$y = - \frac{x \cdot 2}{7} - 3 (\frac{2}{7})$

Step 3.5.2.1.3.2

Combine $- 3$ and $\frac{2}{7}$ .

$y = - \frac{x \cdot 2}{7} + \frac{- 3 \cdot 2}{7}$

Step 3.5.2.1.3.3

Multiply $- 3$ by $2$ .

$y = - \frac{x \cdot 2}{7} + \frac{- 6}{7}$

Step 3.5.2.1.4

Simplify each term.

Tap for more steps...

Step 3.5.2.1.4.1

Move $2$ to the left of $x$ .

$y = - \frac{2 \cdot x}{7} + \frac{- 6}{7}$

Step 3.5.2.1.4.2

Move the negative in front of the fraction.

$y = - \frac{2 x}{7} - \frac{6}{7}$

Step 4

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

$f^{- 1} (x) = - \frac{2 x}{7} - \frac{6}{7}$

Step 5

Verify if $f^{- 1} (x) = - \frac{2 x}{7} - \frac{6}{7}$ is the inverse of $f (x) = - \frac{7}{2} x - 3$ .

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 $f^{- 1} (- \frac{7}{2} x - 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{7}{2} x - 3) = - \frac{2 (- \frac{7}{2} x - 3)}{7} - \frac{6}{7}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{- 2 (- \frac{7}{2} x - 3) - 6}{7}$

Step 5.2.4

Simplify each term.

Tap for more steps...

Step 5.2.4.1

Simplify each term.

Tap for more steps...

Step 5.2.4.1.1

Combine $x$ and $\frac{7}{2}$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{- 2 (- \frac{x \cdot 7}{2} - 3) - 6}{7}$

Step 5.2.4.1.2

Move $7$ to the left of $x$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{- 2 (- \frac{7 x}{2} - 3) - 6}{7}$

Step 5.2.4.2

Apply the distributive property.

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{- 2 (- \frac{7 x}{2}) - 2 \cdot - 3 - 6}{7}$

Step 5.2.4.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.4.3.1

Move the leading negative in $- \frac{7 x}{2}$ into the numerator.

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{- 2 \frac{- 7 x}{2} - 2 \cdot - 3 - 6}{7}$

Step 5.2.4.3.2

Factor $2$ out of $- 2$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{2 (- 1) (\frac{- 7 x}{2}) - 2 \cdot - 3 - 6}{7}$

Step 5.2.4.3.3

Cancel the common factor.

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{2 \cdot (- 1 \frac{- 7 x}{2}) - 2 \cdot - 3 - 6}{7}$

Step 5.2.4.3.4

Rewrite the expression.

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{- 1 (- 7 x) - 2 \cdot - 3 - 6}{7}$

Step 5.2.4.4

Multiply $- 7$ by $- 1$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{7 x - 2 \cdot - 3 - 6}{7}$

Step 5.2.4.5

Multiply $- 2$ by $- 3$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{7 x + 6 - 6}{7}$

Step 5.2.5

Simplify terms.

Tap for more steps...

Step 5.2.5.1

Combine the opposite terms in $7 x + 6 - 6$ .

Tap for more steps...

Step 5.2.5.1.1

Subtract $6$ from $6$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{7 x + 0}{7}$

Step 5.2.5.1.2

Add $7 x$ and $0$ .

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{7 x}{7}$

Step 5.2.5.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (- \frac{7}{2} x - 3) = \frac{7 x}{7}$

Step 5.2.5.2.2

Divide $x$ by $1$ .

$f^{- 1} (- \frac{7}{2} x - 3) = x$

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 $f (- \frac{2 x}{7} - \frac{6}{7})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = - \frac{7}{2} \cdot (- \frac{2 x}{7} - \frac{6}{7}) - 3$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Apply the distributive property.

$f (- \frac{2 x}{7} - \frac{6}{7}) = - \frac{7}{2} \cdot (- \frac{2 x}{7}) - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.2

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.2.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{- 7}{2} \cdot (- \frac{2 x}{7}) - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.2.2

Move the leading negative in $- \frac{2 x}{7}$ into the numerator.

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{- 7}{2} \cdot \frac{- 2 x}{7} - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.2.3

Factor $7$ out of $- 7$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{7 (- 1)}{2} \cdot \frac{- 2 x}{7} - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.2.4

Cancel the common factor.

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{7 \cdot - 1}{2} \cdot \frac{- 2 x}{7} - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.2.5

Rewrite the expression.

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{- 1}{2} \cdot (- 2 x) - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.3.3.1

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

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{- 1}{2} \cdot (2 (- x)) - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.3.2

Cancel the common factor.

$f (- \frac{2 x}{7} - \frac{6}{7}) = \frac{- 1}{2} \cdot (2 (- x)) - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.3.3

Rewrite the expression.

$f (- \frac{2 x}{7} - \frac{6}{7}) = - 1 (- x) - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.4

Multiply $- 1$ by $- 1$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = 1 x - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.5

Multiply $x$ by $1$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = x - \frac{7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.6

Cancel the common factor of $7$ .

Tap for more steps...

Step 5.3.3.6.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{- 7}{2} \cdot (- \frac{6}{7}) - 3$

Step 5.3.3.6.2

Move the leading negative in $- \frac{6}{7}$ into the numerator.

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{- 7}{2} \cdot \frac{- 6}{7} - 3$

Step 5.3.3.6.3

Factor $7$ out of $- 7$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{7 (- 1)}{2} \cdot \frac{- 6}{7} - 3$

Step 5.3.3.6.4

Cancel the common factor.

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{7 \cdot - 1}{2} \cdot \frac{- 6}{7} - 3$

Step 5.3.3.6.5

Rewrite the expression.

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{- 1}{2} \cdot - 6 - 3$

Step 5.3.3.7

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.3.3.7.1

Factor $2$ out of $- 6$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{- 1}{2} \cdot (2 (- 3)) - 3$

Step 5.3.3.7.2

Cancel the common factor.

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + \frac{- 1}{2} \cdot (2 \cdot - 3) - 3$

Step 5.3.3.7.3

Rewrite the expression.

$f (- \frac{2 x}{7} - \frac{6}{7}) = x - 1 \cdot - 3 - 3$

Step 5.3.3.8

Multiply $- 1$ by $- 3$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + 3 - 3$

Step 5.3.4

Combine the opposite terms in $x + 3 - 3$ .

Tap for more steps...

Step 5.3.4.1

Subtract $3$ from $3$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (- \frac{2 x}{7} - \frac{6}{7}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{2 x}{7} - \frac{6}{7}$ is the inverse of $f (x) = - \frac{7}{2} x - 3$ .

$f^{- 1} (x) = - \frac{2 x}{7} - \frac{6}{7}$