Math Problem Statement

Find the domain and range and sketch the domain of the following function: f(x, y) = sqrt(x^2 - 2y).

Solution

The function provided in the image is:

f(x,y)=x22yf(x, y) = \sqrt{x^2 - 2y}

1. Domain:

The domain of a function refers to all the possible input values (in this case, xx and yy) for which the function is defined.

For a square root function, the expression inside the square root must be non-negative. Therefore, we need:

x22y0x^2 - 2y \geq 0

Solving this inequality:

x22yx^2 \geq 2y

yx22y \leq \frac{x^2}{2}

Thus, the domain of the function is all pairs (x,y)(x, y) where yx22y \leq \frac{x^2}{2}.

2. Range:

The range of a function is the set of all possible output values.

Since the function is a square root, the output of the square root is always non-negative. Therefore, the range of the function is:

f(x,y)0f(x, y) \geq 0

Thus, the range of the function is all non-negative real numbers [0,)[0, \infty).

3. Sketching the Domain:

To sketch the domain, you would plot the region yx22y \leq \frac{x^2}{2} in the xyxy-plane. This represents the area below or on the parabola y=x22y = \frac{x^2}{2}.

  • The curve y=x22y = \frac{x^2}{2} is a parabola that opens upwards.
  • The domain includes all points (x,y)(x, y) below or on this curve.

Do you need further explanation or help sketching the domain?

Related Questions:

  1. How do you determine the domain of other types of functions (like logarithmic or rational functions)?
  2. What happens if the expression inside the square root was x2+2yx^2 + 2y instead of x22yx^2 - 2y?
  3. Can a function have an empty domain? What would that mean?
  4. How do you determine the range for functions other than square roots?
  5. How do transformations (like scaling or translating) affect the domain and range of functions?

Tip:

Always check for restrictions in a function (such as square roots or divisions) when determining the domain!

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Math Problem Analysis

Mathematical Concepts

Functions
Inequalities
Domain and Range
Square Root Function

Formulas

x^2 - 2y >= 0
y <= x^2 / 2

Theorems

The Square Root Function Theorem
Domain and Range Determination

Suitable Grade Level

Grade 11-12