Algebra Examples

$f^{- 1} (x) = 4 (20 - x)$

Step 1

Simplify $f^{- 1} x$ .

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Step 1.1

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

$\frac{1}{f} x = 4 (20 - x)$

Step 1.2

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

$\frac{x}{f} = 4 (20 - x)$

Step 2

Simplify $4 (20 - x)$ .

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Step 2.1

Apply the distributive property.

$\frac{x}{f} = 4 \cdot 20 + 4 (- x)$

Step 2.2

Multiply.

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Step 2.2.1

Multiply $4$ by $20$ .

$\frac{x}{f} = 80 + 4 (- x)$

Step 2.2.2

Multiply $- 1$ by $4$ .

$\frac{x}{f} = 80 - 4 x$

Step 3

Add $4 x$ to both sides of the equation.

$\frac{x}{f} + 4 x = 80$

Step 4

Find the LCD of the terms in the equation.

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Step 4.1

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 4.2

The LCM of one and any expression is the expression.

$f$

Step 5

Multiply each term in $\frac{x}{f} + 4 x = 80$ by $f$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{x}{f} + 4 x = 80$ by $f$ .

$\frac{x}{f} f + 4 x f = 80 f$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $f$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{x}{f} f + 4 x f = 80 f$

Step 5.2.1.2

Rewrite the expression.

$x + 4 x f = 80 f$

Step 6

Solve the equation.

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Step 6.1

Factor $x$ out of $x + 4 x f$ .

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Step 6.1.1

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

$x + 4 x f = 80 f$

Step 6.1.2

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

$x \cdot 1 + 4 x f = 80 f$

Step 6.1.3

Factor $x$ out of $4 x f$ .

$x \cdot 1 + x (4 f) = 80 f$

Step 6.1.4

Factor $x$ out of $x \cdot 1 + x (4 f)$ .

$x (1 + 4 f) = 80 f$

Step 6.2

Divide each term in $x (1 + 4 f) = 80 f$ by $1 + 4 f$ and simplify.

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Step 6.2.1

Divide each term in $x (1 + 4 f) = 80 f$ by $1 + 4 f$ .

$\frac{x (1 + 4 f)}{1 + 4 f} = \frac{80 f}{1 + 4 f}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $1 + 4 f$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{x (1 + 4 f)}{1 + 4 f} = \frac{80 f}{1 + 4 f}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{80 f}{1 + 4 f}$

Step 7

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.

$f (f) = \frac{80 f}{1 + 4 f}$