Algebra Examples

$f^{- 1} (0.8)$

Step 1

Interchange the variables.

$f = y^{- 1} (0.8)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y^{- 1} \cdot 0.8 = f$ .

$y^{- 1} \cdot 0.8 = f$

Step 2.2

Divide each term in $y^{- 1} \cdot 0.8 = f$ by $0.8$ and simplify.

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Step 2.2.1

Divide each term in $y^{- 1} \cdot 0.8 = f$ by $0.8$ .

$\frac{y^{- 1} \cdot 0.8}{0.8} = \frac{f}{0.8}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Move $y^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{0.8}{0.8 y} = \frac{f}{0.8}$

Step 2.2.2.2

Cancel the common factor of $0.8$ .

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Step 2.2.2.2.1

Cancel the common factor.

$\frac{0.8}{0.8 y} = \frac{f}{0.8}$

Step 2.2.2.2.2

Rewrite the expression.

$\frac{1}{y} = \frac{f}{0.8}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply by $1$ .

$\frac{1}{y} = \frac{1 f}{0.8}$

Step 2.2.3.2

Factor $0.8$ out of $0.8$ .

$\frac{1}{y} = \frac{1 f}{0.8 (1)}$

Step 2.2.3.3

Separate fractions.

$\frac{1}{y} = \frac{1}{0.8} \cdot \frac{f}{1}$

Step 2.2.3.4

Divide $1$ by $0.8$ .

$\frac{1}{y} = 1.25 \frac{f}{1}$

Step 2.2.3.5

Divide $f$ by $1$ .

$\frac{1}{y} = 1.25 f$

Step 2.3

Find the LCD of the terms in the equation.

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Step 2.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1$

Step 2.3.2

The LCM of one and any expression is the expression.

$y$

Step 2.4

Multiply each term in $\frac{1}{y} = 1.25 f$ by $y$ to eliminate the fractions.

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Step 2.4.1

Multiply each term in $\frac{1}{y} = 1.25 f$ by $y$ .

$\frac{1}{y} y = 1.25 f y$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $y$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{1}{y} y = 1.25 f y$

Step 2.4.2.1.2

Rewrite the expression.

$1 = 1.25 f y$

Step 2.5

Solve the equation.

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Step 2.5.1

Rewrite the equation as $1.25 f y = 1$ .

$1.25 f y = 1$

Step 2.5.2

Divide each term in $1.25 f y = 1$ by $1.25 f$ and simplify.

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Step 2.5.2.1

Divide each term in $1.25 f y = 1$ by $1.25 f$ .

$\frac{1.25 f y}{1.25 f} = \frac{1}{1.25 f}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $1.25$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{1.25 f y}{1.25 f} = \frac{1}{1.25 f}$

Step 2.5.2.2.1.2

Rewrite the expression.

$\frac{f y}{f} = \frac{1}{1.25 f}$

Step 2.5.2.2.2

Cancel the common factor of $f$ .

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Step 2.5.2.2.2.1

Cancel the common factor.

$\frac{f y}{f} = \frac{1}{1.25 f}$

Step 2.5.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{1}{1.25 f}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Rewrite $1$ as $1 (1)$ .

$y = \frac{1 (1)}{1.25 f}$

Step 2.5.2.3.2

Factor $1.25$ out of $1.25 f$ .

$y = \frac{1 (1)}{1.25 (f)}$

Step 2.5.2.3.3

Separate fractions.

$y = \frac{1}{1.25} \cdot \frac{1}{f}$

Step 2.5.2.3.4

Divide $1$ by $1.25$ .

$y = 0.8 \frac{1}{f}$

Step 2.5.2.3.5

Combine $0.8$ and $\frac{1}{f}$ .

$y = \frac{0.8}{f}$

Step 3

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

$f^{- 1} (f) = \frac{0.8}{f}$

Step 4

Verify if $f^{- 1} (f) = \frac{0.8}{f}$ is the inverse of $f (f) = f^{- 1} (0.8)$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (f^{- 1} (0.8))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (f^{- 1} (0.8)) = \frac{0.8}{f^{- 1} (0.8)}$

Step 4.2.3

Move $f^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$f^{- 1} (f^{- 1} (0.8)) = \frac{0.8 f}{0.8}$

Step 4.2.4

Cancel the common factor of $0.8$ .

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Step 4.2.4.1

Cancel the common factor.

$f^{- 1} (f^{- 1} (0.8)) = \frac{0.8 f}{0.8}$

Step 4.2.4.2

Divide $f$ by $1$ .

$f^{- 1} (f^{- 1} (0.8)) = f$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{0.8}{f})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{0.8}{f}) = {(\frac{0.8}{f})}^{- 1} (0.8)$

Step 4.3.3

Change the sign of the exponent by rewriting the base as its reciprocal.

$f (\frac{0.8}{f}) = \frac{f}{0.8} \cdot 0.8$

Step 4.3.4

Cancel the common factor of $0.8$ .

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Step 4.3.4.1

Cancel the common factor.

$f (\frac{0.8}{f}) = \frac{f}{0.8} \cdot 0.8$

Step 4.3.4.2

Rewrite the expression.

$f (\frac{0.8}{f}) = f$

Step 4.4

Since $f^{- 1} (f (f)) = f$ and $f (f^{- 1} (f)) = f$ , then $f^{- 1} (f) = \frac{0.8}{f}$ is the inverse of $f (f) = f^{- 1} (0.8)$ .

$f^{- 1} (f) = \frac{0.8}{f}$