Find the Inverse 3/4x

$\frac{3}{4} x$

Step 1

Interchange the variables.

$x = \frac{3}{4} y$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{3}{4} y = x$ .

$\frac{3}{4} y = x$

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} (\frac{3}{4} y) = \frac{4}{3} x$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{4}{3} (\frac{3}{4} y)$ .

Combine $\frac{3}{4}$ and $y$ .

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} x$

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 y}{4} = \frac{4}{3} x$

Rewrite the expression.

$\frac{1}{3} (3 y) = \frac{4}{3} x$

Cancel the common factor of $3$ .

Factor $3$ out of $3 y$ .

$\frac{1}{3} (3 (y)) = \frac{4}{3} x$

Cancel the common factor.

$\frac{1}{3} (3 y) = \frac{4}{3} x$

Rewrite the expression.

$y = \frac{4}{3} x$

Simplify the right side.

Combine $\frac{4}{3}$ and $x$ .

$y = \frac{4 x}{3}$

Step 3

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

$f^{- 1} (x) = \frac{4 x}{3}$

Step 4

Verify if $f^{- 1} (x) = \frac{4 x}{3}$ is the inverse of $f (x) = \frac{3}{4} x$ .

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 $f^{- 1} (\frac{3}{4} x)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{3}{4} x) = \frac{4 (\frac{3}{4} x)}{3}$

Simplify the numerator.

Combine $4$ and $\frac{3}{4}$ .

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{4 \cdot 3}{4} x}{3}$

Combine $\frac{4 \cdot 3}{4}$ and $x$ .

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{4 \cdot (3 x)}{4}}{3}$

Multiply $4$ by $3$ .

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{12 x}{4}}{3}$

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{12 x}{4}$ by cancelling the common factors.

Factor $4$ out of $12 x$ .

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{4 (3 x)}{4}}{3}$

Factor $4$ out of $4$ .

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{4 (3 x)}{4 (1)}}{3}$

Cancel the common factor.

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{4 (3 x)}{4 \cdot 1}}{3}$

Rewrite the expression.

$f^{- 1} (\frac{3}{4} x) = \frac{\frac{3 x}{1}}{3}$

Divide $3 x$ by $1$ .

$f^{- 1} (\frac{3}{4} x) = \frac{3 x}{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{3}{4} x) = \frac{3 x}{3}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{3}{4} x) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{4 x}{3})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{4 x}{3}) = \frac{3}{4} \cdot (\frac{4 x}{3})$

Cancel the common factor of $3$ .

Cancel the common factor.

$f (\frac{4 x}{3}) = \frac{3}{4} \cdot \frac{4 x}{3}$

Rewrite the expression.

$f (\frac{4 x}{3}) = \frac{1}{4} \cdot (4 x)$

Cancel the common factor of $4$ .

Factor $4$ out of $4 x$ .

$f (\frac{4 x}{3}) = \frac{1}{4} \cdot (4 (x))$

Cancel the common factor.

$f (\frac{4 x}{3}) = \frac{1}{4} \cdot (4 x)$

Rewrite the expression.

$f (\frac{4 x}{3}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{4 x}{3}$ is the inverse of $f (x) = \frac{3}{4} x$ .

$f^{- 1} (x) = \frac{4 x}{3}$

Find the Inverse 3/4x

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