Find the Inverse f(x)=8x-4

$f (x) = 8 x - 4$

Step 1

Write $f (x) = 8 x - 4$ as an equation.

$y = 8 x - 4$

Step 2

Interchange the variables.

$x = 8 y - 4$

Step 3

Solve for $y$ .

Rewrite the equation as $8 y - 4 = x$ .

$8 y - 4 = x$

Add $4$ to both sides of the equation.

$8 y = x + 4$

Divide each term in $8 y = x + 4$ by $8$ and simplify.

Divide each term in $8 y = x + 4$ by $8$ .

$\frac{8 y}{8} = \frac{x}{8} + \frac{4}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 y}{8} = \frac{x}{8} + \frac{4}{8}$

Divide $y$ by $1$ .

$y = \frac{x}{8} + \frac{4}{8}$

Simplify the right side.

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$y = \frac{x}{8} + \frac{4 (1)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$y = \frac{x}{8} + \frac{4 \cdot 1}{4 \cdot 2}$

Cancel the common factor.

$y = \frac{x}{8} + \frac{4 \cdot 1}{4 \cdot 2}$

Rewrite the expression.

$y = \frac{x}{8} + \frac{1}{2}$

Step 4

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

$f^{- 1} (x) = \frac{x}{8} + \frac{1}{2}$

Step 5

Verify if $f^{- 1} (x) = \frac{x}{8} + \frac{1}{2}$ is the inverse of $f (x) = 8 x - 4$ .

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 $f^{- 1} (8 x - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (8 x - 4) = \frac{8 x - 4}{8} + \frac{1}{2}$

Cancel the common factor of $8 x - 4$ and $8$ .

Factor $4$ out of $8 x$ .

$f^{- 1} (8 x - 4) = \frac{4 (2 x) - 4}{8} + \frac{1}{2}$

Factor $4$ out of $- 4$ .

$f^{- 1} (8 x - 4) = \frac{4 (2 x) + 4 (- 1)}{8} + \frac{1}{2}$

Factor $4$ out of $4 (2 x) + 4 (- 1)$ .

$f^{- 1} (8 x - 4) = \frac{4 (2 x - 1)}{8} + \frac{1}{2}$

Cancel the common factors.

Factor $4$ out of $8$ .

$f^{- 1} (8 x - 4) = \frac{4 (2 x - 1)}{4 (2)} + \frac{1}{2}$

Cancel the common factor.

$f^{- 1} (8 x - 4) = \frac{4 (2 x - 1)}{4 \cdot 2} + \frac{1}{2}$

Rewrite the expression.

$f^{- 1} (8 x - 4) = \frac{2 x - 1}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$f^{- 1} (8 x - 4) = \frac{2 x - 1 + 1}{2}$

Combine the opposite terms in $2 x - 1 + 1$ .

Add $- 1$ and $1$ .

$f^{- 1} (8 x - 4) = \frac{2 x + 0}{2}$

Add $2 x$ and $0$ .

$f^{- 1} (8 x - 4) = \frac{2 x}{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (8 x - 4) = \frac{2 x}{2}$

Divide $x$ by $1$ .

$f^{- 1} (8 x - 4) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x}{8} + \frac{1}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{8} + \frac{1}{2}) = 8 (\frac{x}{8} + \frac{1}{2}) - 4$

Simplify each term.

Apply the distributive property.

$f (\frac{x}{8} + \frac{1}{2}) = 8 (\frac{x}{8}) + 8 (\frac{1}{2}) - 4$

Cancel the common factor of $8$ .

Cancel the common factor.

$f (\frac{x}{8} + \frac{1}{2}) = 8 (\frac{x}{8}) + 8 (\frac{1}{2}) - 4$

Rewrite the expression.

$f (\frac{x}{8} + \frac{1}{2}) = x + 8 (\frac{1}{2}) - 4$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$f (\frac{x}{8} + \frac{1}{2}) = x + 2 (4) (\frac{1}{2}) - 4$

Cancel the common factor.

$f (\frac{x}{8} + \frac{1}{2}) = x + 2 \cdot (4 (\frac{1}{2})) - 4$

Rewrite the expression.

$f (\frac{x}{8} + \frac{1}{2}) = x + 4 - 4$

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

Subtract $4$ from $4$ .

$f (\frac{x}{8} + \frac{1}{2}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x}{8} + \frac{1}{2}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x}{8} + \frac{1}{2}$ is the inverse of $f (x) = 8 x - 4$ .

$f^{- 1} (x) = \frac{x}{8} + \frac{1}{2}$

Find the Inverse f(x)=8x-4

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