Question

\(\mathrm{A}=[-1,1], \mathrm{B}=[0,1], \mathrm{C}=[-1,0]\) \(\mathrm{S}_{1}=\left\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x} \in \mathrm{A}, \mathrm{y} \in \mathrm{A}\right\\}\) \(\mathrm{S}_{2}=\left\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x} \in \mathrm{A}, \mathrm{y} \in \mathrm{B}\right\\}\) \(S_{3}=\left\\{(x, y) / x^{2}+y^{2}=1, x \in A, y \in C\right\\}\) \(\mathrm{S}_{4}=\left\\{(\mathrm{x}, \mathrm{y}) / \mathrm{x}^{2}+\mathrm{y}^{2}=1, \mathrm{x} \in \mathrm{B}, \mathrm{y} \in \mathrm{C}\right\\}\) then (a) \(\mathrm{S}_{1}\) is not a graph of a function (b) \(\mathrm{S}_{2}\) is not a graph of a function (c) \(S_{3}\) is not a graph of a function (d) \(\mathrm{S}_{4}\) is not a graph of a function

Short answer

The short answer is: (a) S1 is a graph of a function. (b) S2 is a graph of a function. (c) S3 is not a graph of a function. (d) S4 is a graph of a function.

Step-by-step solution

Identify the domains and ranges for each relation

For each set S1, S2, S3, and S4, let's identify the domains (x-values) and ranges (y-values) of the relations as defined by different sets A, B, and C. S1: Domain (A) = [-1,1], Range (A) = [-1,1] S2: Domain (A) = [-1,1], Range (B) = [0,1] S3: Domain (A) = [-1,1], Range (C) = [-1,0] S4: Domain (B) = [0,1], Range (C) = [-1,0]

Determine whether each relation is a graph of a function

Now, we will check if the given relation is a graph of a function or not using the graphical definition of a function (each x-value related to exactly one y-value) with the equation \(x^2 + y^2 = 1\). S1: If we consider an x-value in A, let's say x = 1, then the two possible y-values related to this x-value are y = 0 and y = 0, which corresponds to exactly one y-value. Similarly, for any x-value in A, there will be only one corresponding y-value due to the equation's symmetry. Therefore, S1 is a graph of a function. S2: If we consider an x-value in A, let's say x = 1, then the two possible y-values related to this x-value are y = 0 and y = 0, which corresponds to exactly one y-value. Similarly, for any x-value in A, there will be only one corresponding y-value in the range B [0, 1], so S2 is a graph of a function. S3: Consider the equation \(x^2 + y^2 = 1\) and the set A [-1,1]. If we look at any x-value in A, let's say x = 0.5, then the two possible y-values related to this x-value are y = \(+\sqrt{1-x^2}\) and y = \(-\sqrt{1-x^2}\). Since both solutions for y falls into different values within the range C [-1,0], S3 is not a graph of a function. S4: Similarly, for S4, consider any x-value in B, let's say x = 0.5, then the two possible y-values related to this x-value is y = \(+\sqrt{1-x^2}\) and y = \(-\sqrt{1-x^2}\). Both solutions for y are not in the range C [-1,0], so there is no value of x in the domain B [0,1] that has two different y-values in the range C [-1,0]. Thus, S4 is a graph of a function. Based on this analysis, the correct answer is: (a) S1 is a graph of a function. (b) S2 is a graph of a function. (c) S3 is not a graph of a function. (d) S4 is a graph of a function.

Key concepts

Graph of a Function

Understanding the graph of a function is key to interpreting relationships between variables. A function can be visualized by plotting its graph on a coordinate plane. For a relation to be considered a function, each x-value (input) must correspond to exactly one y-value (output). This can be tested using the vertical line test. If a vertical line touches the graph at more than one point, then it's not a function.

• Example: In the relation given by the equation \(x^2 + y^2 = 1\), we examine if each x-value has a unique y-value. If it does, then the graph represents a function.
• The equation \(x^2 + y^2 = 1\) is that of a circle, which often means multiple y-values for a single x-value, violating the function criteria.

Recognizing function graphs helps in predicting the behavior of mathematical models.

Domain and Range

The domain and range of a function are fundamental concepts that describe the set of possible inputs and outputs.

The domain refers to all possible x-values that the function can accept without causing any undefined behavior. The range is the set of all possible y-values the function can output based on the domain.

• For example: The domain for set \(A = [-1, 1]\) reflects all x-values in the interval from -1 to 1.
• Similarly, for set \(B = [0, 1]\), the range would be y-values between 0 and 1.

Carefully determining the domain and range is crucial for the correct analysis of the function, ensuring we only consider valid values that satisfy the relation.

Coordinate Geometry

Coordinate Geometry is essential in visualizing equations and relations, connecting algebraic expressions to geometric shapes. It deals with plotting points, lines, and curves on a coordinate plane.

• The equation \(x^2 + y^2 = 1\) traces out a circle centered at the origin with radius 1.
• The symmetrical nature of a circle helps in understanding how x-values relate to y-values, leading to insights on whether it's a function or not.

Using coordinate geometry techniques, one can easily discern the structure and characteristics of mathematical objects, forming the backbone of analyzing functions and relations.

Quadratic Equations

Quadratic equations, such as \(x^2 + y^2 = 1\), form the basis of many complex mathematical concepts. Although typically quadratic equations take the form \(ax^2 + bx + c = 0\), this circle equation represents a special case.

• Quadratic equations are utilized to define curves like parabolas, ellipses, and in this case, a circle.
• These curves can be symmetric, offering two distinct y-values for a given x-value, as seen in our analysis of the circle equation.

Understanding these equations involves recognizing their geometric interpretations, helping decode the nature of the relation between the variables.