Precalculus Examples

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

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 $4$ .

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

Step 3.3

Simplify the left side.

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Step 3.3.1

Cancel the common factor of $4$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

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

Step 3.4

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 3.5

Simplify the exponent.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.5.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.5.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.5.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.1.1.1.2.1

Cancel the common factor.

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

Step 3.5.1.1.1.2.2

Rewrite the expression.

$y^{1} = {(4 x)}^{2}$

Step 3.5.1.1.2

Simplify.

$y = {(4 x)}^{2}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify ${(4 x)}^{2}$ .

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Step 3.5.2.1.1

Apply the product rule to $4 x$ .

$y = 4^{2} x^{2}$

Step 3.5.2.1.2

Raise $4$ to the power of $2$ .

$y = 16 x^{2}$

Step 4

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

$f^{- 1} (x) = 16 x^{2}$

Step 5

Verify if $f^{- 1} (x) = 16 x^{2}$ 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 $f^{- 1}$ .

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 {(\frac{x^{\frac{1}{2}}}{4})}^{2}$

Step 5.2.3

Apply the product rule to $\frac{x^{\frac{1}{2}}}{4}$ .

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{{(x^{\frac{1}{2}})}^{2}}{4^{2}})$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 5.2.4.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{x^{\frac{1}{2} \cdot 2}}{4^{2}})$

Step 5.2.4.1.2

Cancel the common factor of $2$ .

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Step 5.2.4.1.2.1

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{x^{\frac{1}{2} \cdot 2}}{4^{2}})$

Step 5.2.4.1.2.2

Rewrite the expression.

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{x}{4^{2}})$

Step 5.2.4.2

Simplify.

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{x}{4^{2}})$

Step 5.2.5

Reduce the expression by cancelling the common factors.

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Step 5.2.5.1

Raise $4$ to the power of $2$ .

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{x}{16})$

Step 5.2.5.2

Cancel the common factor of $16$ .

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Step 5.2.5.2.1

Cancel the common factor.

$f^{- 1} (\frac{x^{\frac{1}{2}}}{4}) = 16 (\frac{x}{16})$

Step 5.2.5.2.2

Rewrite the expression.

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

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 $f (16 x^{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (16 x^{2}) = \frac{{(16 x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3

Simplify the numerator.

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Step 5.3.3.1

Apply the product rule to $16 x^{2}$ .

$f (16 x^{2}) = \frac{16^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3.2

Rewrite $16$ as $4^{2}$ .

$f (16 x^{2}) = \frac{{(4^{2})}^{\frac{1}{2}} {(x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (16 x^{2}) = \frac{4^{2 (\frac{1}{2})} {(x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3.4

Cancel the common factor of $2$ .

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Step 5.3.3.4.1

Cancel the common factor.

$f (16 x^{2}) = \frac{4^{2 (\frac{1}{2})} {(x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3.4.2

Rewrite the expression.

$f (16 x^{2}) = \frac{4 {(x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3.5

Evaluate the exponent.

$f (16 x^{2}) = \frac{4 {(x^{2})}^{\frac{1}{2}}}{4}$

Step 5.3.3.6

Multiply the exponents in ${(x^{2})}^{\frac{1}{2}}$ .

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Step 5.3.3.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (16 x^{2}) = \frac{4 x^{2 (\frac{1}{2})}}{4}$

Step 5.3.3.6.2

Cancel the common factor of $2$ .

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Step 5.3.3.6.2.1

Cancel the common factor.

$f (16 x^{2}) = \frac{4 x^{2 (\frac{1}{2})}}{4}$

Step 5.3.3.6.2.2

Rewrite the expression.

$f (16 x^{2}) = \frac{4 x}{4}$

Step 5.3.3.7

Simplify.

$f (16 x^{2}) = \frac{4 x}{4}$

Step 5.3.4

Cancel the common factor of $4$ .

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Step 5.3.4.1

Cancel the common factor.

$f (16 x^{2}) = \frac{4 x}{4}$

Step 5.3.4.2

Divide $x$ by $1$ .

$f (16 x^{2}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 16 x^{2}$ is the inverse of $f (x) = \frac{x^{\frac{1}{2}}}{4}$ .

$f^{- 1} (x) = 16 x^{2}$