Find the Inverse f(x)=(x+7)/5

$f (x) = \frac{x + 7}{5}$

Step 1

Write $f (x) = \frac{x + 7}{5}$ as an equation.

$y = \frac{x + 7}{5}$

Step 2

Interchange the variables.

$x = \frac{y + 7}{5}$

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{y + 7}{5} = x$ .

$\frac{y + 7}{5} = x$

Multiply both sides of the equation by $5$ .

$5 \frac{y + 7}{5} = 5 x$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{y + 7}{5} = 5 x$

Rewrite the expression.

$y + 7 = 5 x$

Subtract $7$ from both sides of the equation.

$y = 5 x - 7$

Step 4

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

$f^{- 1} (x) = 5 x - 7$

Step 5

Verify if $f^{- 1} (x) = 5 x - 7$ is the inverse of $f (x) = \frac{x + 7}{5}$ .

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 $f^{- 1} (\frac{x + 7}{5})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x + 7}{5}) = 5 (\frac{x + 7}{5}) - 7$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (\frac{x + 7}{5}) = 5 (\frac{x + 7}{5}) - 7$

Rewrite the expression.

$f^{- 1} (\frac{x + 7}{5}) = x + 7 - 7$

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

Subtract $7$ from $7$ .

$f^{- 1} (\frac{x + 7}{5}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x + 7}{5}) = x$

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

Set up the composite result function.

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

Evaluate $f (5 x - 7)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (5 x - 7) = \frac{(5 x - 7) + 7}{5}$

Simplify the numerator.

Add $- 7$ and $7$ .

$f (5 x - 7) = \frac{5 x + 0}{5}$

Add $5 x$ and $0$ .

$f (5 x - 7) = \frac{5 x}{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$f (5 x - 7) = \frac{5 x}{5}$

Divide $x$ by $1$ .

$f (5 x - 7) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 5 x - 7$ is the inverse of $f (x) = \frac{x + 7}{5}$ .

$f^{- 1} (x) = 5 x - 7$

Find the Inverse f(x)=(x+7)/5

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