Finite Math Examples

$f (x) = k x^{- 2}$

Step 1

Write $f (x) = k x^{- 2}$ as an equation.

$y = k x^{- 2}$

Step 2

Interchange the variables.

$k = y x^{- 2}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $y x^{- 2} = k$ .

$y x^{- 2} = k$

Step 3.2

Divide each term in $y x^{- 2} = k$ by $x^{- 2}$ and simplify.

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Step 3.2.1

Divide each term in $y x^{- 2} = k$ by $x^{- 2}$ .

$\frac{y x^{- 2}}{x^{- 2}} = \frac{k}{x^{- 2}}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x^{- 2}$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{y x^{- 2}}{x^{- 2}} = \frac{k}{x^{- 2}}$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{k}{x^{- 2}}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move $x^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$y = k x^{2}$

Step 4

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

$f^{- 1} (k) = k x^{2}$

Step 5

Verify if $f^{- 1} (k) = k x^{2}$ is the inverse of $f (k) = k x^{- 2}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (k x^{- 2})$ by substituting in the value of $f$ into $f^{- 1}$ .

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

Step 5.2.3

Multiply $x^{- 2}$ by $x^{2}$ by adding the exponents.

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Step 5.2.3.1

Move $x^{2}$ .

$f^{- 1} (k x^{- 2}) = k (x^{2} x^{- 2})$

Step 5.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (k x^{- 2}) = k x^{2 - 2}$

Step 5.2.3.3

Subtract $2$ from $2$ .

$f^{- 1} (k x^{- 2}) = k x^{0}$

Step 5.2.4

Simplify $k x^{0}$ .

$f^{- 1} (k x^{- 2}) = k$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (k x^{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (k x^{2}) = (k x^{2}) \cdot x^{- 2}$

Step 5.3.3

Multiply $x^{2}$ by $x^{- 2}$ by adding the exponents.

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Step 5.3.3.1

Move $x^{- 2}$ .

$f (k x^{2}) = k (x^{- 2} x^{2})$

Step 5.3.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (k x^{2}) = k x^{- 2 + 2}$

Step 5.3.3.3

Add $- 2$ and $2$ .

$f (k x^{2}) = k x^{0}$

Step 5.3.4

Simplify $k x^{0}$ .

$f (k x^{2}) = k$

Step 5.4

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

$f^{- 1} (k) = k x^{2}$