Find the Inverse f(x)=1/(4x)

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

Find the LCD of the terms in the equation.

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

$4 y, 1$

The LCM of one and any expression is the expression.

$4 y$

Multiply each term in $\frac{1}{4 y} = x$ by $4 y$ to eliminate the fractions.

Multiply each term in $\frac{1}{4 y} = x$ by $4 y$ .

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

Simplify the left side.

Rewrite using the commutative property of multiplication.

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

Cancel the common factor of $4$ .

Factor $4$ out of $4 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $y$ .

Cancel the common factor.

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

Rewrite the expression.

$1 = x (4 y)$

Simplify the right side.

Rewrite using the commutative property of multiplication.

$1 = 4 x y$

Solve the equation.

Rewrite the equation as $4 x y = 1$ .

$4 x y = 1$

Divide each term in $4 x y = 1$ by $4 x$ and simplify.

Divide each term in $4 x y = 1$ by $4 x$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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

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

Set up the composite result function.

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

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

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

Combine $4$ and $\frac{1}{4 x}$ .

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

Reduce the expression $\frac{4}{4 x}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Combine $4$ and $\frac{1}{4 x}$ .

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

Reduce the expression $\frac{4}{4 x}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $x$ by $1$ .

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

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

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

Find the Inverse f(x)=1/(4x)

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