Algebra Examples

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

Step 1

Find the range of the given function.

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Step 1.1

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

$[- 7, \infty)$

Step 1.2

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

$y \geq - 7$

Step 2

Find the inverse.

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Step 2.1

Interchange the variables.

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

Step 2.2

Solve for $y$ .

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Step 2.2.1

Rewrite the equation as $2 y^{2} - 8 y + 1 = x$ .

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

Step 2.2.2

Subtract $x$ from both sides of the equation.

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

Step 2.2.3

Use the quadratic formula to find the solutions.

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

Step 2.2.4

Substitute the values $a = 2$ , $b = - 8$ , and $c = 1 - x$ into the quadratic formula and solve for $y$ .

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

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 $- 8$ to the power of $2$ .

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

Step 2.2.5.1.2

Multiply $- 4$ by $2$ .

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

Step 2.2.5.1.3

Apply the distributive property.

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

Step 2.2.5.1.4

Multiply $- 8$ by $1$ .

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

Step 2.2.5.1.5

Multiply $- 1$ by $- 8$ .

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

Step 2.2.5.1.6

Subtract $8$ from $64$ .

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

Step 2.2.5.1.7

Factor $8$ out of $56 + 8 x$ .

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Step 2.2.5.1.7.1

Factor $8$ out of $56$ .

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

Step 2.2.5.1.7.2

Factor $8$ out of $8 \cdot 7 + 8 x$ .

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

Step 2.2.5.1.8

Rewrite $8 (7 + x)$ as $2^{2} \cdot (2 (7 + x))$ .

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Step 2.2.5.1.8.1

Factor $4$ out of $8$ .

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

Step 2.2.5.1.8.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.2.5.1.8.3

Add parentheses.

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

Step 2.2.5.1.9

Pull terms out from under the radical.

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

Step 2.2.5.2

Multiply $2$ by $2$ .

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

Step 2.2.5.3

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

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

Step 2.2.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

Raise $- 8$ to the power of $2$ .

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

Step 2.2.6.1.2

Multiply $- 4$ by $2$ .

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

Step 2.2.6.1.3

Apply the distributive property.

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

Step 2.2.6.1.4

Multiply $- 8$ by $1$ .

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

Step 2.2.6.1.5

Multiply $- 1$ by $- 8$ .

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

Step 2.2.6.1.6

Subtract $8$ from $64$ .

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

Step 2.2.6.1.7

Factor $8$ out of $56 + 8 x$ .

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Step 2.2.6.1.7.1

Factor $8$ out of $56$ .

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

Step 2.2.6.1.7.2

Factor $8$ out of $8 \cdot 7 + 8 x$ .

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

Step 2.2.6.1.8

Rewrite $8 (7 + x)$ as $2^{2} \cdot (2 (7 + x))$ .

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Step 2.2.6.1.8.1

Factor $4$ out of $8$ .

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

Step 2.2.6.1.8.2

Rewrite $4$ as $2^{2}$ .

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

Step 2.2.6.1.8.3

Add parentheses.

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

Step 2.2.6.1.9

Pull terms out from under the radical.

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

Step 2.2.6.2

Multiply $2$ by $2$ .

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

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