Which of the following graphs represent valid functions? This is a common question in mathematics, particularly in the study of algebra and calculus. Understanding the characteristics of a valid function is crucial for solving problems and analyzing mathematical relationships. In this article, we will explore the criteria for determining whether a graph represents a valid function and discuss some examples to illustrate the concepts.
A valid function is a relation between two sets of numbers, where each input value (x) corresponds to exactly one output value (y). In other words, for every x-value, there should be a unique y-value. This concept is often referred to as the vertical line test. If a vertical line intersects the graph at more than one point, then the graph does not represent a valid function.
Let’s examine some examples to understand this concept better.
Example 1:
Consider the graph of the function f(x) = x^2. This graph is a parabola that opens upward. To determine if it represents a valid function, we can apply the vertical line test. If we draw a vertical line at any point on the graph, it will intersect the parabola at only one point. Therefore, the graph of f(x) = x^2 represents a valid function.
Example 2:
Now, let’s consider the graph of the function g(x) = |x|. This graph is a V-shaped curve that is symmetric about the y-axis. To determine if it represents a valid function, we can again apply the vertical line test. In this case, if we draw a vertical line at x = 0, it will intersect the graph at two points (0, 0) and (0, 0). This violates the vertical line test, so the graph of g(x) = |x| does not represent a valid function.
Example 3:
Consider the graph of the function h(x) = x^3. This graph is a cubic curve that passes through the origin. To determine if it represents a valid function, we can apply the vertical line test. As with the first example, if we draw a vertical line at any point on the graph, it will intersect the curve at only one point. Therefore, the graph of h(x) = x^3 represents a valid function.
In conclusion, to determine if a graph represents a valid function, we must apply the vertical line test. If a vertical line intersects the graph at more than one point, then the graph does not represent a valid function. By understanding the criteria for a valid function, we can analyze and solve mathematical problems more effectively.
