Find the Inverse f(x)=2x^2-8x+1 , x>=2

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Algebra

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

Find the range of the given function.

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

$[- 7, \infty)$

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

$y \geq - 7$

Find the inverse.

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Interchange the variables.

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

Solve for $y$ .

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

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

Subtract $x$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

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Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

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

Apply the distributive property.

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

Multiply $- 8$ by $1$ .

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

Multiply $- 1$ by $- 8$ .

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

Subtract $8$ from $64$ .

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

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

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Factor $8$ out of $56$ .

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

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

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

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

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Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

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

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

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

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Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

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

Apply the distributive property.

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

Multiply $- 8$ by $1$ .

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

Multiply $- 1$ by $- 8$ .

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

Subtract $8$ from $64$ .

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

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

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Factor $8$ out of $56$ .

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

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

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

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

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Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

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

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

Change the $\pm$ to $+$ .

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

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

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Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

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

Apply the distributive property.

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

Multiply $- 8$ by $1$ .

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

Multiply $- 1$ by $- 8$ .

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

Subtract $8$ from $64$ .

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

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

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Factor $8$ out of $56$ .

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

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

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

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

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Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

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

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

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

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

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

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

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

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

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

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Name

Name	six hundred eleven million two hundred twenty-six thousand forty-five

Interesting facts

611226045 has 16 divisors, whose sum is 1303948992
The reverse of 611226045 is 540622116
Previous prime number is 5

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 27
Digital Root 9

Name

Name	one billion nine hundred ninety-five million six hundred fourteen thousand nine hundred seventy-eight

Interesting facts

1995614978 has 8 divisors, whose sum is 3421054272
The reverse of 1995614978 is 8794165991
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 59
Digital Root 5

Name

Name	one billion three hundred eighty-five million two hundred fifty-six thousand one hundred six

Interesting facts

1385256106 has 8 divisors, whose sum is 2237721444
The reverse of 1385256106 is 6016525831
Previous prime number is 13

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 37
Digital Root 1

Factor x^3+1

Multiply (7- square root of 2)(8+ square root of 2)

Find the Inverse Function f(x)=x^3-5

Solve by Factoring x^3+3x^2-4x-12=0

Rationalize the Denominator 4/(9- square root of 7)