Algebra Examples

$f^{- 1} (\frac{3 + x}{x - 2}) = x + 1$

Step 1

Simplify $f^{- 1} (\frac{3 + x}{x - 2})$ .

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{f} \cdot \frac{3 + x}{x - 2} = x + 1$

Step 1.2

Multiply $\frac{1}{f}$ by $\frac{3 + x}{x - 2}$ .

$\frac{3 + x}{f (x - 2)} = x + 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$f (x - 2), 1, 1$

Step 2.2

The LCM of one and any expression is the expression.

$f (x - 2)$

Step 3

Multiply each term in $\frac{3 + x}{f (x - 2)} = x + 1$ by $f (x - 2)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{3 + x}{f (x - 2)} = x + 1$ by $f (x - 2)$ .

$\frac{3 + x}{f (x - 2)} (f (x - 2)) = x (f (x - 2)) + 1 (f (x - 2))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $f (x - 2)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 + x}{f (x - 2)} (f (x - 2)) = x (f (x - 2)) + 1 (f (x - 2))$

Step 3.2.1.2

Rewrite the expression.

$3 + x = x (f (x - 2)) + 1 (f (x - 2))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Apply the distributive property.

$3 + x = x (f x + f \cdot - 2) + 1 (f (x - 2))$

Step 3.3.1.2

Move $- 2$ to the left of $f$ .

$3 + x = x (f x - 2 \cdot f) + 1 (f (x - 2))$

Step 3.3.1.3

Apply the distributive property.

$3 + x = x (f x) + x (- 2 f) + 1 (f (x - 2))$

Step 3.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.1.4.1

Move $x$ .

$3 + x = x \cdot x f + x (- 2 f) + 1 (f (x - 2))$

Step 3.3.1.4.2

Multiply $x$ by $x$ .

$3 + x = x^{2} f + x (- 2 f) + 1 (f (x - 2))$

Step 3.3.1.5

Rewrite using the commutative property of multiplication.

$3 + x = x^{2} f - 2 x f + 1 (f (x - 2))$

Step 3.3.1.6

Multiply $f (x - 2)$ by $1$ .

$3 + x = x^{2} f - 2 x f + f (x - 2)$

Step 3.3.1.7

Apply the distributive property.

$3 + x = x^{2} f - 2 x f + f x + f \cdot - 2$

Step 3.3.1.8

Move $- 2$ to the left of $f$ .

$3 + x = x^{2} f - 2 x f + f x - 2 f$

Step 3.3.2

Add $- 2 x f$ and $f x$ .

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Step 3.3.2.1

Move $x$ .

$3 + x = x^{2} f - 2 f x + f x - 2 f$

Step 3.3.2.2

Add $- 2 f x$ and $f x$ .

$3 + x = x^{2} f - f x - 2 f$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $x^{2} f - f x - 2 f = 3 + x$ .

$x^{2} f - f x - 2 f = 3 + x$

Step 4.2

Factor $f$ out of $x^{2} f - f x - 2 f$ .

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Step 4.2.1

Factor $f$ out of $x^{2} f$ .

$f x^{2} - f x - 2 f = 3 + x$

Step 4.2.2

Factor $f$ out of $- f x$ .

$f x^{2} + f (- 1 x) - 2 f = 3 + x$

Step 4.2.3

Factor $f$ out of $- 2 f$ .

$f x^{2} + f (- 1 x) + f \cdot - 2 = 3 + x$

Step 4.2.4

Factor $f$ out of $f x^{2} + f (- 1 x)$ .

$f (x^{2} - 1 x) + f \cdot - 2 = 3 + x$

Step 4.2.5

Factor $f$ out of $f (x^{2} - 1 x) + f \cdot - 2$ .

$f (x^{2} - 1 x - 2) = 3 + x$

Step 4.3

Factor.

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Step 4.3.1

Factor $x^{2} - 1 x - 2$ using the AC method.

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Step 4.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 4.3.1.2

Write the factored form using these integers.

$f ((x - 2) (x + 1)) = 3 + x$

Step 4.3.2

Remove unnecessary parentheses.

$f (x - 2) (x + 1) = 3 + x$

Step 4.4

Divide each term in $f (x - 2) (x + 1) = 3 + x$ by $(x - 2) (x + 1)$ and simplify.

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Step 4.4.1

Divide each term in $f (x - 2) (x + 1) = 3 + x$ by $(x - 2) (x + 1)$ .

$\frac{f (x - 2) (x + 1)}{(x - 2) (x + 1)} = \frac{3}{(x - 2) (x + 1)} + \frac{x}{(x - 2) (x + 1)}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $x - 2$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{f (x - 2) (x + 1)}{(x - 2) (x + 1)} = \frac{3}{(x - 2) (x + 1)} + \frac{x}{(x - 2) (x + 1)}$

Step 4.4.2.1.2

Rewrite the expression.

$\frac{f (x + 1)}{x + 1} = \frac{3}{(x - 2) (x + 1)} + \frac{x}{(x - 2) (x + 1)}$

Step 4.4.2.2

Cancel the common factor of $x + 1$ .

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Step 4.4.2.2.1

Cancel the common factor.

$\frac{f (x + 1)}{x + 1} = \frac{3}{(x - 2) (x + 1)} + \frac{x}{(x - 2) (x + 1)}$

Step 4.4.2.2.2

Divide $f$ by $1$ .

$f = \frac{3}{(x - 2) (x + 1)} + \frac{x}{(x - 2) (x + 1)}$