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anonymous

  • one year ago

How do I algebraically find the domain of sqrt (1 - x^2)? (i'll re-post as an equation) I know the answer is [-1,1]. But how do I go about solving it algebraically.

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  1. anonymous
    • one year ago
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    \[\sqrt{1-x^2}\]

  2. Vocaloid
    • one year ago
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    set the expression underneath the radical greater or equal to 0 \[1-x^{2} \ge 0 \]

  3. anonymous
    • one year ago
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    Is what I have below correct? \[1 - x^2 \ge 0 \] \[1 \ge x^2\] \[\sqrt{1}\ge \sqrt{x^2}\] \[\pm1\ge x\]

  4. Vocaloid
    • one year ago
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    trying to come up with a good way to explain this, sorry \[1 \ge x^{2}\] results in \[-1 \le x \le 1\]

  5. anonymous
    • one year ago
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    Thanks

  6. SolomonZelman
    • one year ago
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    Yes, you are correct. √1=√(x²) 1=|x| x=±1

  7. mathstudent55
    • one year ago
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    The radicand must be non-negative, or \(1 - x^2 \ge 0\) To solve this inequality, factor the left side and solve it as an equation: \((1 + x)(1 - x) = 0\) \(1 + x = 0\) or \(1 - x = 0\) \(x = -1\) or \(x = 1\) Now you have 2 points of interest. These two points of interest divide the number line into 3 regions (not counting the points themselves). |dw:1437618309443:dw|

  8. mathstudent55
    • one year ago
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    Since the inequality has the symbol \(\ge\) which contains =, the points of interest are part of the solution, so you put closed dots on them. |dw:1437618438257:dw|

  9. mathstudent55
    • one year ago
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    Now you test each of the three regions to see which ones are part of the solution. To test each region, choose a point in teat region and test is. If the point works, then the entire region works. Let's test -2 \(1 - (-2)^2 \ge 0\) \(1 - 4 \ge 0\) \(-3 \ge 0\) is false, so left of -1 is not part of the solution. Now let's test 0 \(1 - 0^2 \ge 0\) \(1 - 0 \ge 0\) \(1 \ge 0\) is true, so between -1 and 1 is part of the solution. Now let's test 2 \(1 - ^2 \ge 0\) \(1 - 4 \ge 0\) \(-3 \ge 0\) is false, so to the right of 1 is not part of the solution. |dw:1437618822904:dw|

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