Algebra Examples

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

Step 1

Simplify $f^{- 1} x$ .

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Step 1.1

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

$\frac{1}{f} x = 2 x + 1$

Step 1.2

Combine $\frac{1}{f}$ and $x$ .

$\frac{x}{f} = 2 x + 1$

Step 2

Subtract $2 x$ from both sides of the equation.

$\frac{x}{f} - 2 x = 1$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$f, 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$f$

Step 4

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

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Step 4.1

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

$\frac{x}{f} f - 2 x f = 1 f$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $f$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x}{f} f - 2 x f = 1 f$

Step 4.2.1.2

Rewrite the expression.

$x - 2 x f = 1 f$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $f$ by $1$ .

$x - 2 x f = f$

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

Raise $x$ to the power of $1$ .

$x - 2 x f = f$

Step 5.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - 2 x f = f$

Step 5.1.3

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

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

Step 5.1.4

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

$x (1 - 2 f) = f$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $1 - 2 f$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{f}{1 - 2 f}$

Step 6

To rewrite as a function of $f$ , write the equation so that $x$ is by itself on one side of the equal sign and an expression involving only $f$ is on the other side.

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