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x=2|y|+5
Find four points contained in the inverse. Express your values as an integer or simplified fraction.

x=2y+5x=2|y|+5\newlineFind four points contained in the inverse. Express your values as an integer or simplified fraction.

Full solution

Q. x=2y+5x=2|y|+5\newlineFind four points contained in the inverse. Express your values as an integer or simplified fraction.
  1. Express yy in terms of xx: To find the inverse of the function, we first need to express yy in terms of xx. This means we need to solve the equation for yy.
  2. Isolate y term: Subtract 55 from both sides of the equation to isolate the term with yy on one side.\newlinex5=2yx - 5 = 2|y|
  3. Solve for y|y|: Now, divide both sides by 22 to solve for y|y|.(x5)/2=y(x - 5) / 2 = |y|
  4. Consider two cases: Since |y|\
  5. Find points for inverse: Now we need to find four points that are contained in the inverse. We can choose arbitrary values for \(x and solve for yy in both cases.\newlineLet's choose x=7x = 7, x=9x = 9, x=11x = 11, and x=13x = 13 as our xx-values.
  6. Point 11: x=7x=7: For x=7x = 7:
    Case 11: y=(75)2=22=1y = \frac{(7 - 5)}{2} = \frac{2}{2} = 1
    Case 22: y=(75)2=22=1y = -\frac{(7 - 5)}{2} = -\frac{2}{2} = -1
    So, the points are (7,1)(7, 1) and (7,1)(7, -1).
  7. Point 22: x=9x=9: For x=9x = 9:
    Case 11: y=(95)2=42=2y = \frac{(9 - 5)}{2} = \frac{4}{2} = 2
    Case 22: y=(95)2=42=2y = -\frac{(9 - 5)}{2} = -\frac{4}{2} = -2
    So, the points are (9,2)(9, 2) and (9,2)(9, -2).
  8. Point 33: x=11x=11: For x=11x = 11:
    Case 11: y=(115)2=62=3y = \frac{(11 - 5)}{2} = \frac{6}{2} = 3
    Case 22: y=(115)2=62=3y = -\frac{(11 - 5)}{2} = -\frac{6}{2} = -3
    So, the points are (11,3)(11, 3) and (11,3)(11, -3).
  9. Point 44: x=13x=13: For x=13x = 13:
    Case 11: y=(135)2=82=4y = \frac{(13 - 5)}{2} = \frac{8}{2} = 4
    Case 22: y=(135)2=82=4y = -\frac{(13 - 5)}{2} = -\frac{8}{2} = -4
    So, the points are (13,4)(13, 4) and (13,4)(13, -4).

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