Precalculus Examples

$g (x) = \frac{3}{x} + 1$

Step 1

Write $g (x) = \frac{3}{x} + 1$ as an equation.

$y = \frac{3}{x} + 1$

Step 2

Interchange the variables.

$x = \frac{3}{y} + 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\frac{3}{y} + 1 = x$ .

$\frac{3}{y} + 1 = x$

Step 3.2

Subtract $1$ from both sides of the equation.

$\frac{3}{y} = x - 1$

Step 3.3

Find the LCD of the terms in the equation.

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Step 3.3.1

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

$y, 1, 1$

Step 3.3.2

The LCM of one and any expression is the expression.

$y$

Step 3.4

Multiply each term in $\frac{3}{y} = x - 1$ by $y$ to eliminate the fractions.

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Step 3.4.1

Multiply each term in $\frac{3}{y} = x - 1$ by $y$ .

$\frac{3}{y} y = x y - y$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $y$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{3}{y} y = x y - y$

Step 3.4.2.1.2

Rewrite the expression.

$3 = x y - y$

Step 3.5

Solve the equation.

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Step 3.5.1

Rewrite the equation as $x y - y = 3$ .

$x y - y = 3$

Step 3.5.2

Factor $y$ out of $x y - y$ .

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Step 3.5.2.1

Factor $y$ out of $x y$ .

$y x - y = 3$

Step 3.5.2.2

Factor $y$ out of $- y$ .

$y x + y \cdot - 1 = 3$

Step 3.5.2.3

Factor $y$ out of $y x + y \cdot - 1$ .

$y (x - 1) = 3$

Step 3.5.3

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

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Step 3.5.3.1

Divide each term in $y (x - 1) = 3$ by $x - 1$ .

$\frac{y (x - 1)}{x - 1} = \frac{3}{x - 1}$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $x - 1$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{y (x - 1)}{x - 1} = \frac{3}{x - 1}$

Step 3.5.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{3}{x - 1}$

Step 4

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

$g^{- 1} (x) = \frac{3}{x - 1}$

Step 5

Verify if $g^{- 1} (x) = \frac{3}{x - 1}$ is the inverse of $g (x) = \frac{3}{x} + 1$ .

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Step 5.1

To verify the inverse, check if $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ .

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $g^{- 1} (\frac{3}{x} + 1)$ by substituting in the value of $g$ into $g^{- 1}$ .

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

Step 5.2.3

Simplify the denominator.

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Step 5.2.3.1

Subtract $1$ from $1$ .

$g^{- 1} (\frac{3}{x} + 1) = \frac{3}{\frac{3}{x} + 0}$

Step 5.2.3.2

Add $\frac{3}{x}$ and $0$ .

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

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.2.5

Cancel the common factor of $3$ .

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Step 5.2.5.1

Cancel the common factor.

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

Step 5.2.5.2

Rewrite the expression.

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $g (\frac{3}{x - 1})$ by substituting in the value of $g^{- 1}$ into $g$ .

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

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.3.2

Cancel the common factor of $3$ .

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Step 5.3.3.2.1

Cancel the common factor.

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

Step 5.3.3.2.2

Rewrite the expression.

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

Step 5.3.4

Combine the opposite terms in $x - 1 + 1$ .

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Step 5.3.4.1

Add $- 1$ and $1$ .

$g (\frac{3}{x - 1}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$g (\frac{3}{x - 1}) = x$

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = \frac{3}{x - 1}$ is the inverse of $g (x) = \frac{3}{x} + 1$ .

$g^{- 1} (x) = \frac{3}{x - 1}$