Finite Math Examples

$f (x) = 1.5 x^{2} - 4$

Step 1

Write $f (x) = 1.5 x^{2} - 4$ as an equation.

$y = 1.5 x^{2} - 4$

Step 2

Interchange the variables.

$x = 1.5 y^{2} - 4$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $1.5 y^{2} - 4 = x$ .

$1.5 y^{2} - 4 = x$

Step 3.2

Add $4$ to both sides of the equation.

$1.5 y^{2} = x + 4$

Step 3.3

Divide each term in $1.5 y^{2} = x + 4$ by $1.5$ and simplify.

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Step 3.3.1

Divide each term in $1.5 y^{2} = x + 4$ by $1.5$ .

$\frac{1.5 y^{2}}{1.5} = \frac{x}{1.5} + \frac{4}{1.5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $1.5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1.5 y^{2}}{1.5} = \frac{x}{1.5} + \frac{4}{1.5}$

Step 3.3.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{x}{1.5} + \frac{4}{1.5}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply by $1$ .

$y^{2} = \frac{1 x}{1.5} + \frac{4}{1.5}$

Step 3.3.3.1.2

Factor $1.5$ out of $1.5$ .

$y^{2} = \frac{1 x}{1.5 (1)} + \frac{4}{1.5}$

Step 3.3.3.1.3

Separate fractions.

$y^{2} = \frac{1}{1.5} \cdot \frac{x}{1} + \frac{4}{1.5}$

Step 3.3.3.1.4

Divide $1$ by $1.5$ .

$y^{2} = 0.\overline{6} \frac{x}{1} + \frac{4}{1.5}$

Step 3.3.3.1.5

Divide $x$ by $1$ .

$y^{2} = 0.\overline{6} x + \frac{4}{1.5}$

Step 3.3.3.1.6

Divide $4$ by $1.5$ .

$y^{2} = 0.\overline{6} x + 2.\overline{6}$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{0.\overline{6} x + 2.\overline{6}}$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{0.\overline{6} x + 2.\overline{6}}$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{0.\overline{6} x + 2.\overline{6}}$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{0.\overline{6} x + 2.\overline{6}}$

$y = - \sqrt{0.\overline{6} x + 2.\overline{6}}$

$y = \sqrt{0.\overline{6} x + 2.\overline{6}}$

$y = - \sqrt{0.\overline{6} x + 2.\overline{6}}$

$y = \sqrt{0.\overline{6} x + 2.\overline{6}}$

$y = - \sqrt{0.\overline{6} x + 2.\overline{6}}$

Step 4

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

$f^{- 1} (x) = \sqrt{0.\overline{6} x + 2.\overline{6}}, - \sqrt{0.\overline{6} x + 2.\overline{6}}$

Step 5

Verify if $f^{- 1} (x) = \sqrt{0.\overline{6} x + 2.\overline{6}}, - \sqrt{0.\overline{6} x + 2.\overline{6}}$ is the inverse of $f (x) = 1.5 x^{2} - 4$ .

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Step 5.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 1.5 x^{2} - 4$ and $f^{- 1} (x) = \sqrt{0.\overline{6} x + 2.\overline{6}}, - \sqrt{0.\overline{6} x + 2.\overline{6}}$ and compare them.

Step 5.2

Find the range of $f (x) = 1.5 x^{2} - 4$ .

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Step 5.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- 4, \infty)$

Step 5.3

Find the domain of $\sqrt{0.\overline{6} x + 2.\overline{6}}$ .

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Step 5.3.1

Set the radicand in $\sqrt{0.\overline{6} x + 2.\overline{6}}$ greater than or equal to $0$ to find where the expression is defined.

$0.\overline{6} x + 2.\overline{6} \geq 0$

Step 5.3.2

Solve for $x$ .

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Step 5.3.2.1

Subtract $2.\overline{6}$ from both sides of the inequality.

$0.\overline{6} x \geq - 2.\overline{6}$

Step 5.3.2.2

Divide each term in $0.\overline{6} x \geq - 2.\overline{6}$ by $0.\overline{6}$ and simplify.

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Step 5.3.2.2.1

Divide each term in $0.\overline{6} x \geq - 2.\overline{6}$ by $0.\overline{6}$ .

$\frac{0.\overline{6} x}{0.\overline{6}} \geq \frac{- 2.\overline{6}}{0.\overline{6}}$

Step 5.3.2.2.2

Simplify the left side.

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Step 5.3.2.2.2.1

Cancel the common factor of $0.\overline{6}$ .

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Step 5.3.2.2.2.1.1

Cancel the common factor.

$\frac{0.\overline{6} x}{0.\overline{6}} \geq \frac{- 2.\overline{6}}{0.\overline{6}}$

Step 5.3.2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 2.\overline{6}}{0.\overline{6}}$

Step 5.3.2.2.3

Simplify the right side.

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Step 5.3.2.2.3.1

Divide $- 2.\overline{6}$ by $0.\overline{6}$ .

$x \geq - 4$

Step 5.3.3

The domain is all values of $x$ that make the expression defined.

$[- 4, \infty)$

Step 5.4

Since the domain of $f^{- 1} (x) = \sqrt{0.\overline{6} x + 2.\overline{6}}, - \sqrt{0.\overline{6} x + 2.\overline{6}}$ is not equal to the range of $f (x) = 1.5 x^{2} - 4$ , then $f^{- 1} (x) = \sqrt{0.\overline{6} x + 2.\overline{6}}, - \sqrt{0.\overline{6} x + 2.\overline{6}}$ is not an inverse of $f (x) = 1.5 x^{2} - 4$ .

There is no inverse

Step 6