Find the Inverse f(x)=4/3x-7

$f (x) = \frac{4}{3} x - 7$

Step 1

Write $f (x) = \frac{4}{3} x - 7$ as an equation.

$y = \frac{4}{3} x - 7$

Step 2

Interchange the variables.

$x = \frac{4}{3} y - 7$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{4}{3} y - 7 = x$ .

$\frac{4}{3} y - 7 = x$

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

$\frac{4 y}{3} - 7 = x$

Add $7$ to both sides of the equation.

$\frac{4 y}{3} = x + 7$

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

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x + 7)$

Simplify both sides of the equation.

Simplify the left side.

Simplify $\frac{3}{4} \cdot \frac{4 y}{3}$ .

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 y}{3} = \frac{3}{4} (x + 7)$

Rewrite the expression.

$\frac{1}{4} (4 y) = \frac{3}{4} (x + 7)$

Cancel the common factor of $4$ .

Factor $4$ out of $4 y$ .

$\frac{1}{4} (4 (y)) = \frac{3}{4} (x + 7)$

Cancel the common factor.

$\frac{1}{4} (4 y) = \frac{3}{4} (x + 7)$

Rewrite the expression.

$y = \frac{3}{4} (x + 7)$

Simplify the right side.

Simplify $\frac{3}{4} (x + 7)$ .

Apply the distributive property.

$y = \frac{3}{4} x + \frac{3}{4} \cdot 7$

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

$y = \frac{3 x}{4} + \frac{3}{4} \cdot 7$

Multiply $\frac{3}{4} \cdot 7$ .

Combine $\frac{3}{4}$ and $7$ .

$y = \frac{3 x}{4} + \frac{3 \cdot 7}{4}$

Multiply $3$ by $7$ .

$y = \frac{3 x}{4} + \frac{21}{4}$

Step 4

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

$f^{- 1} (x) = \frac{3 x}{4} + \frac{21}{4}$

Step 5

Verify if $f^{- 1} (x) = \frac{3 x}{4} + \frac{21}{4}$ is the inverse of $f (x) = \frac{4}{3} x - 7$ .

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 $f^{- 1} (\frac{4}{3} x - 7)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{4}{3} x - 7) = \frac{3 (\frac{4}{3} x - 7)}{4} + \frac{21}{4}$

Combine the numerators over the common denominator.

$f^{- 1} (\frac{4}{3} x - 7) = \frac{3 (\frac{4}{3} x - 7) + 21}{4}$

Simplify each term.

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

$f^{- 1} (\frac{4}{3} x - 7) = \frac{3 (\frac{4 x}{3} - 7) + 21}{4}$

Apply the distributive property.

$f^{- 1} (\frac{4}{3} x - 7) = \frac{3 (\frac{4 x}{3}) + 3 \cdot - 7 + 21}{4}$

Cancel the common factor of $3$ .

Cancel the common factor.

$f^{- 1} (\frac{4}{3} x - 7) = \frac{3 (\frac{4 x}{3}) + 3 \cdot - 7 + 21}{4}$

Rewrite the expression.

$f^{- 1} (\frac{4}{3} x - 7) = \frac{4 x + 3 \cdot - 7 + 21}{4}$

Multiply $3$ by $- 7$ .

$f^{- 1} (\frac{4}{3} x - 7) = \frac{4 x - 21 + 21}{4}$

Simplify terms.

Combine the opposite terms in $4 x - 21 + 21$ .

Add $- 21$ and $21$ .

$f^{- 1} (\frac{4}{3} x - 7) = \frac{4 x + 0}{4}$

Add $4 x$ and $0$ .

$f^{- 1} (\frac{4}{3} x - 7) = \frac{4 x}{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$f^{- 1} (\frac{4}{3} x - 7) = \frac{4 x}{4}$

Divide $x$ by $1$ .

$f^{- 1} (\frac{4}{3} x - 7) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{3 x}{4} + \frac{21}{4})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{3 x}{4} + \frac{21}{4}) = \frac{4}{3} \cdot (\frac{3 x}{4} + \frac{21}{4}) - 7$

Simplify each term.

Apply the distributive property.

$f (\frac{3 x}{4} + \frac{21}{4}) = \frac{4}{3} \cdot \frac{3 x}{4} + \frac{4}{3} \cdot \frac{21}{4} - 7$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{3 x}{4} + \frac{21}{4}) = \frac{4}{3} \cdot \frac{3 x}{4} + \frac{4}{3} \cdot \frac{21}{4} - 7$

Rewrite the expression.

$f (\frac{3 x}{4} + \frac{21}{4}) = \frac{1}{3} \cdot (3 x) + \frac{4}{3} \cdot \frac{21}{4} - 7$

Cancel the common factor of $3$ .

Factor $3$ out of $3 x$ .

$f (\frac{3 x}{4} + \frac{21}{4}) = \frac{1}{3} \cdot (3 (x)) + \frac{4}{3} \cdot \frac{21}{4} - 7$

Cancel the common factor.

$f (\frac{3 x}{4} + \frac{21}{4}) = \frac{1}{3} \cdot (3 x) + \frac{4}{3} \cdot \frac{21}{4} - 7$

Rewrite the expression.

$f (\frac{3 x}{4} + \frac{21}{4}) = x + \frac{4}{3} \cdot \frac{21}{4} - 7$

Cancel the common factor of $4$ .

Cancel the common factor.

$f (\frac{3 x}{4} + \frac{21}{4}) = x + \frac{4}{3} \cdot \frac{21}{4} - 7$

Rewrite the expression.

$f (\frac{3 x}{4} + \frac{21}{4}) = x + \frac{1}{3} \cdot 21 - 7$

Cancel the common factor of $3$ .

Factor $3$ out of $21$ .

$f (\frac{3 x}{4} + \frac{21}{4}) = x + \frac{1}{3} \cdot (3 (7)) - 7$

Cancel the common factor.

$f (\frac{3 x}{4} + \frac{21}{4}) = x + \frac{1}{3} \cdot (3 \cdot 7) - 7$

Rewrite the expression.

$f (\frac{3 x}{4} + \frac{21}{4}) = x + 7 - 7$

Combine the opposite terms in $x + 7 - 7$ .

Subtract $7$ from $7$ .

$f (\frac{3 x}{4} + \frac{21}{4}) = x + 0$

Add $x$ and $0$ .

$f (\frac{3 x}{4} + \frac{21}{4}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{3 x}{4} + \frac{21}{4}$ is the inverse of $f (x) = \frac{4}{3} x - 7$ .

$f^{- 1} (x) = \frac{3 x}{4} + \frac{21}{4}$

Find the Inverse f(x)=4/3x-7

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