Algebra Examples

$f (x) = \frac{x \frac{1}{2}}{4}$

Step 1

Write $f (x) = \frac{x \frac{1}{2}}{4}$ as an equation.

$y = \frac{x \frac{1}{2}}{4}$

Step 2

Interchange the variables.

$x = \frac{y (\frac{1}{2})}{4}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{y (\frac{1}{2})}{4} = x$ .

$\frac{y (\frac{1}{2})}{4} = x$

Step 3.2

Multiply both sides of the equation by $\frac{4}{\frac{1}{2}}$ .

$\frac{4}{\frac{1}{2}} \cdot \frac{y \frac{1}{2}}{4} = \frac{4}{\frac{1}{2}} x$

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 $\frac{4}{\frac{1}{2}} \cdot \frac{y \frac{1}{2}}{4}$ .

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Step 3.3.1.1.1

Cancel the common factor of $4$ .

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Step 3.3.1.1.1.1

Cancel the common factor.

$\frac{4}{\frac{1}{2}} \cdot \frac{y \frac{1}{2}}{4} = \frac{4}{\frac{1}{2}} x$

Step 3.3.1.1.1.2

Rewrite the expression.

$\frac{1}{\frac{1}{2}} (y \frac{1}{2}) = \frac{4}{\frac{1}{2}} x$

Step 3.3.1.1.2

Cancel the common factor of $\frac{1}{2}$ .

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Step 3.3.1.1.2.1

Factor $\frac{1}{2}$ out of $y \frac{1}{2}$ .

$\frac{1}{\frac{1}{2}} (\frac{1}{2} y) = \frac{4}{\frac{1}{2}} x$

Step 3.3.1.1.2.2

Cancel the common factor.

$\frac{1}{\frac{1}{2}} (\frac{1}{2} y) = \frac{4}{\frac{1}{2}} x$

Step 3.3.1.1.2.3

Rewrite the expression.

$y = \frac{4}{\frac{1}{2}} x$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\frac{4}{\frac{1}{2}} x$ .

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Step 3.3.2.1.1

Multiply the numerator by the reciprocal of the denominator.

$y = 4 \cdot 2 x$

Step 3.3.2.1.2

Multiply $4$ by $2$ .

$y = 8 x$

Step 4

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

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

Step 5

Verify if $f^{- 1} (x) = 8 x$ is the inverse of $f (x) = \frac{x (\frac{1}{2})}{4}$ .

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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 $f^{- 1} (\frac{x (\frac{1}{2})}{4})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = 8 (\frac{x (\frac{1}{2})}{4})$

Step 5.2.3

Combine $x$ and $\frac{1}{2}$ .

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = 8 (\frac{\frac{x}{2}}{4})$

Step 5.2.4

Cancel the common factor of $4$ .

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Step 5.2.4.1

Factor $4$ out of $8$ .

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = 4 (2) (\frac{\frac{x}{2}}{4})$

Step 5.2.4.2

Cancel the common factor.

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = 4 \cdot (2 (\frac{\frac{x}{2}}{4}))$

Step 5.2.4.3

Rewrite the expression.

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = 2 (\frac{x}{2})$

Step 5.2.5

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

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = \frac{2 x}{2}$

Step 5.2.6

Cancel the common factor of $2$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = \frac{2 x}{2}$

Step 5.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{x (\frac{1}{2})}{4}) = x$

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

$f (8 x) = \frac{(8 x) (\frac{1}{2})}{4}$

Step 5.3.3

Cancel the common factor of $8$ and $4$ .

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Step 5.3.3.1

Factor $4$ out of $(8 x) (\frac{1}{2})$ .

$f (8 x) = \frac{4 (2 x (\frac{1}{2}))}{4}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $4$ out of $4$ .

$f (8 x) = \frac{4 (2 x (\frac{1}{2}))}{4 (1)}$

Step 5.3.3.2.2

Cancel the common factor.

$f (8 x) = \frac{4 (2 x (\frac{1}{2}))}{4 \cdot 1}$

Step 5.3.3.2.3

Rewrite the expression.

$f (8 x) = \frac{2 x (\frac{1}{2})}{1}$

Step 5.3.3.2.4

Divide $2 x \frac{1}{2}$ by $1$ .

$f (8 x) = 2 x (\frac{1}{2})$

Step 5.3.4

Cancel the common factor of $2$ .

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Step 5.3.4.1

Factor $2$ out of $2 x$ .

$f (8 x) = 2 (x) (\frac{1}{2})$

Step 5.3.4.2

Cancel the common factor.

$f (8 x) = 2 x (\frac{1}{2})$

Step 5.3.4.3

Rewrite the expression.

$f (8 x) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 8 x$ is the inverse of $f (x) = \frac{x (\frac{1}{2})}{4}$ .

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