Algebra Examples

f (x) = 2 x^{2} - 8 x + 1

$f (x) = 2 x^{2} - 8 x + 1$ ,

x \geq 2

$x \geq 2$

Step 1

Find the range of the given function.

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Step 1.1

The range is the set of all valid

y

$y$ values. Use the graph to find the range.

[- 7, \infty)

$[- 7, \infty)$

Step 1.2

Convert

[- 7, \infty)

$[- 7, \infty)$ to an inequality.

y \geq - 7

$y \geq - 7$

y \geq - 7

$y \geq - 7$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

x = 2 y^{2} - 8 y + 1

$x = 2 y^{2} - 8 y + 1$

Step 2.2

Solve for

y

$y$ .

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Step 2.2.1

Rewrite the equation as

2 y^{2} - 8 y + 1 = x

$2 y^{2} - 8 y + 1 = x$ .

2 y^{2} - 8 y + 1 = x

$2 y^{2} - 8 y + 1 = x$

Step 2.2.2

Subtract

x

$x$ from both sides of the equation.

2 y^{2} - 8 y + 1 - x = 0

$2 y^{2} - 8 y + 1 - x = 0$

Step 2.2.3

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2.4

Substitute the values

a = 2

$a = 2$ ,

b = - 8

$b = - 8$ , and

c = 1 - x

$c = 1 - x$ into the quadratic formula and solve for

y

$y$ .

\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (2 \cdot (1 - x))}}{2 \cdot 2}

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (2 \cdot (1 - x))}}{2 \cdot 2}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

Raise

- 8

$- 8$ to the power of

2

$2$ .

y = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot (1 - x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot (1 - x)}}{2 \cdot 2}$

Step 2.2.5.1.2

Multiply

- 4

$- 4$ by

2

$2$ .

y = \frac{8 \pm \sqrt{64 - 8 \cdot (1 - x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 \cdot (1 - x)}}{2 \cdot 2}$

Step 2.2.5.1.3

Apply the distributive property.

y = \frac{8 \pm \sqrt{64 - 8 \cdot 1 - 8 (- x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 \cdot 1 - 8 (- x)}}{2 \cdot 2}$

Step 2.2.5.1.4

Multiply

- 8

$- 8$ by

1

$1$ .

y = \frac{8 \pm \sqrt{64 - 8 - 8 (- x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 - 8 (- x)}}{2 \cdot 2}$

Step 2.2.5.1.5

Multiply

- 1

$- 1$ by

- 8

$- 8$ .

y = \frac{8 \pm \sqrt{64 - 8 + 8 x}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 + 8 x}}{2 \cdot 2}$

Step 2.2.5.1.6

Subtract

8

$8$ from

64

$64$ .

y = \frac{8 \pm \sqrt{56 + 8 x}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{56 + 8 x}}{2 \cdot 2}$

Step 2.2.5.1.7

Factor

8

$8$ out of

56 + 8 x

$56 + 8 x$ .

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Step 2.2.5.1.7.1

Factor

8

$8$ out of

56

$56$ .

y = \frac{8 \pm \sqrt{8 \cdot 7 + 8 x}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{8 \cdot 7 + 8 x}}{2 \cdot 2}$

Step 2.2.5.1.7.2

Factor

8

$8$ out of

8 \cdot 7 + 8 x

$8 \cdot 7 + 8 x$ .

y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}$

y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}$

Step 2.2.5.1.8

Rewrite

8 (7 + x)

$8 (7 + x)$ as

2^{2} \cdot (2 (7 + x))

$2^{2} \cdot (2 (7 + x))$ .

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Step 2.2.5.1.8.1

Factor

4

$4$ out of

8

$8$ .

y = \frac{8 \pm \sqrt{4 (2) (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{4 (2) (7 + x)}}{2 \cdot 2}$

Step 2.2.5.1.8.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

Step 2.2.5.1.8.3

Add parentheses.

y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

Step 2.2.5.1.9

Pull terms out from under the radical.

y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}$

y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}$

Step 2.2.5.2

Multiply

2

$2$ by

2

$2$ .

y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}$

Step 2.2.5.3

Simplify

\frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}

$\frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}$ .

y = \frac{4 \pm \sqrt{2 (7 + x)}}{2}

$y = \frac{4 \pm \sqrt{2 (7 + x)}}{2}$

y = \frac{4 \pm \sqrt{2 (7 + x)}}{2}

$y = \frac{4 \pm \sqrt{2 (7 + x)}}{2}$

Step 2.2.6

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise

- 8

$- 8$ to the power of

2

$2$ .

y = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot (1 - x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot (1 - x)}}{2 \cdot 2}$

Step 2.2.6.1.2

Multiply

- 4

$- 4$ by

2

$2$ .

y = \frac{8 \pm \sqrt{64 - 8 \cdot (1 - x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 \cdot (1 - x)}}{2 \cdot 2}$

Step 2.2.6.1.3

Apply the distributive property.

y = \frac{8 \pm \sqrt{64 - 8 \cdot 1 - 8 (- x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 \cdot 1 - 8 (- x)}}{2 \cdot 2}$

Step 2.2.6.1.4

Multiply

- 8

$- 8$ by

1

$1$ .

y = \frac{8 \pm \sqrt{64 - 8 - 8 (- x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 - 8 (- x)}}{2 \cdot 2}$

Step 2.2.6.1.5

Multiply

- 1

$- 1$ by

- 8

$- 8$ .

y = \frac{8 \pm \sqrt{64 - 8 + 8 x}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{64 - 8 + 8 x}}{2 \cdot 2}$

Step 2.2.6.1.6

Subtract

8

$8$ from

64

$64$ .

y = \frac{8 \pm \sqrt{56 + 8 x}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{56 + 8 x}}{2 \cdot 2}$

Step 2.2.6.1.7

Factor

8

$8$ out of

56 + 8 x

$56 + 8 x$ .

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Step 2.2.6.1.7.1

Factor

8

$8$ out of

56

$56$ .

y = \frac{8 \pm \sqrt{8 \cdot 7 + 8 x}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{8 \cdot 7 + 8 x}}{2 \cdot 2}$

Step 2.2.6.1.7.2

Factor

8

$8$ out of

8 \cdot 7 + 8 x

$8 \cdot 7 + 8 x$ .

y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}$

y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}$

Step 2.2.6.1.8

Rewrite

8 (7 + x)

$8 (7 + x)$ as

2^{2} \cdot (2 (7 + x))

$2^{2} \cdot (2 (7 + x))$ .

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Step 2.2.6.1.8.1

Factor

4

$4$ out of

8

$8$ .

y = \frac{8 \pm \sqrt{4 (2) (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{4 (2) (7 + x)}}{2 \cdot 2}$

Step 2.2.6.1.8.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

Step 2.2.6.1.8.3

Add parentheses.

y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

Step 2.2.6.1.9

Pull terms out from under the radical.

y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}$

y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}$

Step 2.2.6.2

Multiply

2

$2$ by

2

$2$ .

y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}$

Step 2.2.6.3

Simplify

$\frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}$ .

$y = \frac{4 \pm \sqrt{2 (7 + x)}}{2}$

Step 2.2.6.4

Change the

$\pm$ to

$+$ .

$y = \frac{4 + \sqrt{2 (7 + x)}}{2}$

Step 2.2.7

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 2.2.7.1

Simplify the numerator.

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Step 2.2.7.1.1

Raise

$- 8$ to the power of

$2$ .

$y = \frac{8 \pm \sqrt{64 - 4 \cdot 2 \cdot (1 - x)}}{2 \cdot 2}$

Step 2.2.7.1.2

Multiply

$- 4$ by

$2$ .

$y = \frac{8 \pm \sqrt{64 - 8 \cdot (1 - x)}}{2 \cdot 2}$

Step 2.2.7.1.3

Apply the distributive property.

$y = \frac{8 \pm \sqrt{64 - 8 \cdot 1 - 8 (- x)}}{2 \cdot 2}$

Step 2.2.7.1.4

Multiply

$- 8$ by

$1$ .

$y = \frac{8 \pm \sqrt{64 - 8 - 8 (- x)}}{2 \cdot 2}$

Step 2.2.7.1.5

Multiply

$- 1$ by

$- 8$ .

$y = \frac{8 \pm \sqrt{64 - 8 + 8 x}}{2 \cdot 2}$

Step 2.2.7.1.6

Subtract

$8$ from

$64$ .

$y = \frac{8 \pm \sqrt{56 + 8 x}}{2 \cdot 2}$

Step 2.2.7.1.7

Factor

$8$ out of

$56 + 8 x$ .

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Step 2.2.7.1.7.1

Factor

$8$ out of

$56$ .

$y = \frac{8 \pm \sqrt{8 \cdot 7 + 8 x}}{2 \cdot 2}$

Step 2.2.7.1.7.2

Factor

$8$ out of

$8 \cdot 7 + 8 x$ .

$y = \frac{8 \pm \sqrt{8 (7 + x)}}{2 \cdot 2}$

Step 2.2.7.1.8

Rewrite

$8 (7 + x)$ as

$2^{2} \cdot (2 (7 + x))$ .

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Step 2.2.7.1.8.1

Factor

$4$ out of

$8$ .

$y = \frac{8 \pm \sqrt{4 (2) (7 + x)}}{2 \cdot 2}$

Step 2.2.7.1.8.2

Rewrite

$4$ as

$2^{2}$ .

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

Step 2.2.7.1.8.3

Add parentheses.

$y = \frac{8 \pm \sqrt{2^{2} \cdot (2 (7 + x))}}{2 \cdot 2}$

Step 2.2.7.1.9

Pull terms out from under the radical.

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{2 \cdot 2}$

Step 2.2.7.2

Multiply

$2$ by

$2$ .

$y = \frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}$

Step 2.2.7.3

Simplify

$\frac{8 \pm 2 \sqrt{2 (7 + x)}}{4}$ .

$y = \frac{4 \pm \sqrt{2 (7 + x)}}{2}$

Step 2.2.7.4

Change the

$\pm$ to

$-$ .

$y = \frac{4 - \sqrt{2 (7 + x)}}{2}$

Step 2.2.8

The final answer is the combination of both solutions.

$y = \frac{4 + \sqrt{2 (7 + x)}}{2}$

$y = \frac{4 - \sqrt{2 (7 + x)}}{2}$

$y = \frac{4 + \sqrt{2 (7 + x)}}{2}$

$y = \frac{4 - \sqrt{2 (7 + x)}}{2}$

Step 2.3

Replace

$y$ with

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

$f^{- 1} (x) = \frac{4 + \sqrt{2 (7 + x)}}{2}, \frac{4 - \sqrt{2 (7 + x)}}{2}$

Step 3

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = \frac{4 + \sqrt{2 (7 + x)}}{2}, x \geq - 7$

Step 4