Algebra Examples

$f (x) = 2 x + 7$ ; $f^{- 1} (x) = \frac{1}{2} x - \frac{7}{2}$

Step 1

Write $f (x) = 2 x + 7$ as an equation.

$y = 2 x + 7$

Step 2

Interchange the variables.

$x = 2 y + 7$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $2 y + 7 = x$ .

$2 y + 7 = x$

Step 3.2

Subtract $7$ from both sides of the equation.

$2 y = x - 7$

Step 3.3

Divide each term in $2 y = x - 7$ by $2$ and simplify.

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Step 3.3.1

Divide each term in $2 y = x - 7$ by $2$ .

$\frac{2 y}{2} = \frac{x}{2} + \frac{- 7}{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{x}{2} + \frac{- 7}{2}$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x}{2} + \frac{- 7}{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Move the negative in front of the fraction.

$y = \frac{x}{2} - \frac{7}{2}$

Step 4

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

$f^{- 1} (x) = \frac{x}{2} - \frac{7}{2}$

Step 5

Set up the composite result function.

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

Step 6

Evaluate $f (\frac{x}{2} - \frac{7}{2})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x}{2} - \frac{7}{2}) = 2 (\frac{x}{2} - \frac{7}{2}) + 7$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$f (\frac{x}{2} - \frac{7}{2}) = 2 (\frac{x}{2}) + 2 (- \frac{7}{2}) + 7$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

$f (\frac{x}{2} - \frac{7}{2}) = 2 (\frac{x}{2}) + 2 (- \frac{7}{2}) + 7$

Step 7.2.2

Rewrite the expression.

$f (\frac{x}{2} - \frac{7}{2}) = x + 2 (- \frac{7}{2}) + 7$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$f (\frac{x}{2} - \frac{7}{2}) = x + 2 (\frac{- 7}{2}) + 7$

Step 7.3.2

Cancel the common factor.

$f (\frac{x}{2} - \frac{7}{2}) = x + 2 (\frac{- 7}{2}) + 7$

Step 7.3.3

Rewrite the expression.

$f (\frac{x}{2} - \frac{7}{2}) = x - 7 + 7$

Step 8

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

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Step 8.1

Add $- 7$ and $7$ .

$f (\frac{x}{2} - \frac{7}{2}) = x + 0$

Step 8.2

Add $x$ and $0$ .

$f (\frac{x}{2} - \frac{7}{2}) = x$