How do you find the inverse of the function y = 2x^2 + 2?

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To find the inverse of the function y = 2x2 + 2, we need to follow several steps:

  1. Start with the original function: Write the equation with y on one side:

    2. y = 2x2 + 2

  2. Swap y and x: To find the inverse, we interchange x and y in the equation:

    3. x = 2y2 + 2

  3. Isolate y: Our goal is to solve for y. Start by moving the constant term to the other side of the equation:

    4. x - 2 = 2y2

  4. Divide both sides by 2: This simplifies the equation further:

    5. y2 = rac{x - 2}{2}

  5. Take the square root of both sides: Remember to consider both the positive and negative roots:

    6. y = ±√( rac{x - 2}{2})

  6. Write the inverse function: We typically choose the positive root when defining the principal square root. Thus, the inverse function can be denoted as:

    7. f-1(x) = √( rac{x - 2}{2})

  7. Domain and range considerations: Since the original function is a parabola opening upwards, its range is [2, ∞). Therefore, the domain of the inverse function is [2, ∞).

In conclusion, the inverse of the function y = 2x2 + 2 is:

f-1(x) = √( rac{x - 2}{2}), with a domain of [2, ∞).

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