Algebra Examples

$y = 2 x^{2} + 7$

Step 1

Interchange the variables.

$x = 2 y^{2} + 7$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $2 y^{2} + 7 = x$ .

$2 y^{2} + 7 = x$

Step 2.2

Subtract $7$ from both sides of the equation.

$2 y^{2} = x - 7$

Step 2.3

Divide each term in $2 y^{2} = x - 7$ by $2$ and simplify.

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Step 2.3.1

Divide each term in $2 y^{2} = x - 7$ by $2$ .

$\frac{2 y^{2}}{2} = \frac{x}{2} + \frac{- 7}{2}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{2 y^{2}}{2} = \frac{x}{2} + \frac{- 7}{2}$

Step 2.3.2.1.2

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

$y^{2} = \frac{x}{2} + \frac{- 7}{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Move the negative in front of the fraction.

$y^{2} = \frac{x}{2} - \frac{7}{2}$

Step 2.4

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

$y = \pm \sqrt{\frac{x}{2} - \frac{7}{2}}$

Step 2.5

Simplify $\pm \sqrt{\frac{x}{2} - \frac{7}{2}}$ .

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Step 2.5.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x - 7}{2}}$

Step 2.5.2

Rewrite $\sqrt{\frac{x - 7}{2}}$ as $\frac{\sqrt{x - 7}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x - 7}}{\sqrt{2}}$

Step 2.5.3

Multiply $\frac{\sqrt{x - 7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x - 7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.5.4

Combine and simplify the denominator.

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Step 2.5.4.1

Multiply $\frac{\sqrt{x - 7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.5.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.5.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.5.4.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.5.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.5.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.5.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.5.4.6.4

Cancel the common factor of $2$ .

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Step 2.5.4.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.5.4.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{2^{1}}$

Step 2.5.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x - 7} \sqrt{2}}{2}$

Step 2.5.5

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(x - 7) \cdot 2}}{2}$

Step 2.5.6

Reorder factors in $\pm \frac{\sqrt{(x - 7) \cdot 2}}{2}$ .

$y = \pm \frac{\sqrt{2 (x - 7)}}{2}$

Step 2.6

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

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Step 2.6.1

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

$y = \frac{\sqrt{2 (x - 7)}}{2}$

Step 2.6.2

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

$y = - \frac{\sqrt{2 (x - 7)}}{2}$

Step 2.6.3

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

$y = \frac{\sqrt{2 (x - 7)}}{2}$

$y = - \frac{\sqrt{2 (x - 7)}}{2}$

$y = \frac{\sqrt{2 (x - 7)}}{2}$

$y = - \frac{\sqrt{2 (x - 7)}}{2}$

$y = \frac{\sqrt{2 (x - 7)}}{2}$

$y = - \frac{\sqrt{2 (x - 7)}}{2}$

Step 3

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

$f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$

Step 4

Verify if $f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$ is the inverse of $f (x) = 2 x^{2} + 7$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 2 x^{2} + 7$ and $f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$ and compare them.

Step 4.2

Find the range of $f (x) = 2 x^{2} + 7$ .

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Step 4.2.1

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

Interval Notation:

$[7, \infty)$

Step 4.3

Find the domain of $\frac{\sqrt{2 (x - 7)}}{2}$ .

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Step 4.3.1

Set the radicand in $\sqrt{2 (x - 7)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (x - 7) \geq 0$

Step 4.3.2

Solve for $x$ .

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Step 4.3.2.1

Divide each term in $2 (x - 7) \geq 0$ by $2$ and simplify.

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Step 4.3.2.1.1

Divide each term in $2 (x - 7) \geq 0$ by $2$ .

$\frac{2 (x - 7)}{2} \geq \frac{0}{2}$

Step 4.3.2.1.2

Simplify the left side.

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Step 4.3.2.1.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.1.2.1.1

Cancel the common factor.

$\frac{2 (x - 7)}{2} \geq \frac{0}{2}$

Step 4.3.2.1.2.1.2

Divide $x - 7$ by $1$ .

$x - 7 \geq \frac{0}{2}$

Step 4.3.2.1.3

Simplify the right side.

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Step 4.3.2.1.3.1

Divide $0$ by $2$ .

$x - 7 \geq 0$

Step 4.3.2.2

Add $7$ to both sides of the inequality.

$x \geq 7$

Step 4.3.3

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

$[7, \infty)$

Step 4.4

Find the domain of $f (x) = 2 x^{2} + 7$ .

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Step 4.4.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

$(- \infty, \infty)$

Step 4.5

Since the domain of $f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$ is the range of $f (x) = 2 x^{2} + 7$ and the range of $f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$ is the domain of $f (x) = 2 x^{2} + 7$ , then $f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$ is the inverse of $f (x) = 2 x^{2} + 7$ .

$f^{- 1} (x) = \frac{\sqrt{2 (x - 7)}}{2}, - \frac{\sqrt{2 (x - 7)}}{2}$

Step 5