Algebra Examples

$f (x) = \frac{7}{8} x$ , $f^{- 1} (x) = \frac{8}{7} x$

Step 1

Write $f (x) = \frac{7}{8} x$ as an equation.

$y = \frac{7}{8} x$

Step 2

Interchange the variables.

$x = \frac{7}{8} y$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{7}{8} y = x$ .

$\frac{7}{8} y = x$

Step 3.2

Multiply both sides of the equation by $\frac{8}{7}$ .

$\frac{8}{7} (\frac{7}{8} y) = \frac{8}{7} x$

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 $\frac{8}{7} (\frac{7}{8} y)$ .

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Step 3.3.1.1.1

Combine $\frac{7}{8}$ and $y$ .

$\frac{8}{7} \cdot \frac{7 y}{8} = \frac{8}{7} x$

Step 3.3.1.1.2

Cancel the common factor of $8$ .

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Step 3.3.1.1.2.1

Cancel the common factor.

$\frac{8}{7} \cdot \frac{7 y}{8} = \frac{8}{7} x$

Step 3.3.1.1.2.2

Rewrite the expression.

$\frac{1}{7} (7 y) = \frac{8}{7} x$

Step 3.3.1.1.3

Cancel the common factor of $7$ .

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Step 3.3.1.1.3.1

Factor $7$ out of $7 y$ .

$\frac{1}{7} (7 (y)) = \frac{8}{7} x$

Step 3.3.1.1.3.2

Cancel the common factor.

$\frac{1}{7} (7 y) = \frac{8}{7} x$

Step 3.3.1.1.3.3

Rewrite the expression.

$y = \frac{8}{7} x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Combine $\frac{8}{7}$ and $x$ .

$y = \frac{8 x}{7}$

Step 4

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

$f^{- 1} (x) = \frac{8 x}{7}$

Step 5

Set up the composite result function.

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

Step 6

Evaluate $f^{- 1} (\frac{7}{8} x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{7}{8} x) = \frac{8 (\frac{7}{8} x)}{7}$

Step 7

Simplify the numerator.

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Step 7.1

Combine $8$ and $\frac{7}{8}$ .

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{8 \cdot 7}{8} x}{7}$

Step 7.2

Combine $\frac{8 \cdot 7}{8}$ and $x$ .

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{8 \cdot (7 x)}{8}}{7}$

Step 8

Multiply $8$ by $7$ .

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{56 x}{8}}{7}$

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Reduce the expression $\frac{56 x}{8}$ by cancelling the common factors.

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Step 9.1.1

Factor $8$ out of $56 x$ .

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{8 (7 x)}{8}}{7}$

Step 9.1.2

Factor $8$ out of $8$ .

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{8 (7 x)}{8 (1)}}{7}$

Step 9.1.3

Cancel the common factor.

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{8 (7 x)}{8 \cdot 1}}{7}$

Step 9.1.4

Rewrite the expression.

$f^{- 1} (\frac{7}{8} x) = \frac{\frac{7 x}{1}}{7}$

Step 9.2

Divide $7 x$ by $1$ .

$f^{- 1} (\frac{7}{8} x) = \frac{7 x}{7}$

Step 10

Cancel the common factor of $7$ .

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Step 10.1

Cancel the common factor.

$f^{- 1} (\frac{7}{8} x) = \frac{7 x}{7}$

Step 10.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{7}{8} x) = x$