Function And Relation Exercises In Mathematics
In mathematics, understanding the concept of a function is fundamental. A function is a special type of relation where each input (x-value) corresponds to exactly one output (y-value). To determine if a relation is a function, we need to examine its set of ordered pairs. If any input value is associated with more than one output value, then the relation is not a function. Let's delve into the following exercises to solidify this understanding.
1. Identifying Non-Function Relations
The first relation we need to analyze is: {(-5,2),(3,-3),(1,7),(3,0)}. To determine if this relation is a function, we need to examine the input values (x-values) and their corresponding output values (y-values). Notice that the input value 3 appears twice, once with the output -3 and once with the output 0. This means that the input 3 is associated with two different output values.
Therefore, this relation is not a function because the input 3 maps to two different outputs, -3 and 0. This violates the fundamental requirement of a function, where each input must have a unique output. In simpler terms, if you were to plot these points on a graph, you would see a vertical line passing through the point where x=3, intersecting the graph at two different points. This is a visual representation of the vertical line test, which is a quick way to determine if a relation is a function. If any vertical line intersects the graph more than once, the relation is not a function. In this case, the vertical line x=3 would intersect the graph at (3,-3) and (3,0), confirming that it is not a function.
2. Recognizing Function Relations
Now, let's consider the second relation: {(2,7),(-5,20),(-10,20),(-2,10),(1,20)}. Again, we need to examine the input values and their corresponding output values. In this relation, we observe that each input value is unique; that is, no input value is repeated. The input values are 2, -5, -10, -2, and 1, each associated with a single output value. Even though the output value 20 appears three times (corresponding to the inputs -5, -10, and 1), this does not violate the definition of a function. The function definition only restricts multiple outputs for a single input, not multiple inputs for a single output.
Therefore, this relation is a function because each input value has a unique output value. We can confidently say that this relation satisfies the fundamental requirement of a function. In the context of the vertical line test, if you were to plot these points on a graph, no vertical line would intersect the graph more than once. Each vertical line would either intersect the graph at a single point or not at all, confirming that it is a function. The concept of a function is crucial in many areas of mathematics, including calculus, algebra, and analysis, making it essential to understand these fundamental principles.
In addition to determining if a relation is a function, another essential skill in mathematics is evaluating functions. Evaluating a function means finding the output value (y-value) for a given input value (x-value). This process involves substituting the given input value into the function's equation and simplifying the expression to find the corresponding output value. Let's explore how to evaluate functions using the given function f(x) = -3x + 2.
3. Evaluating f(9)
To find the function value f(9), we need to substitute x = 9 into the function's equation. This means replacing the variable x with the number 9 in the expression -3x + 2. The resulting expression is -3(9) + 2. Now, we simplify this expression using the order of operations (PEMDAS/BODMAS). First, we perform the multiplication: -3 * 9 = -27. Then, we perform the addition: -27 + 2 = -25. Therefore, the function value f(9) is equal to -25. This means that when the input is 9, the corresponding output of the function is -25. Understanding how to evaluate functions is crucial for various mathematical applications, including graphing functions, solving equations, and modeling real-world phenomena. The ability to correctly substitute values and simplify expressions is a fundamental skill in mathematics, and mastering it allows for a deeper understanding of functional relationships. Moreover, evaluating functions is not just a mathematical exercise; it has practical applications in various fields such as physics, engineering, and economics, where functions are used to model and analyze real-world situations.
4. Evaluating f(12)
Similarly, to find the function value f(12), we substitute x = 12 into the function's equation f(x) = -3x + 2. Replacing x with 12 gives us the expression -3(12) + 2. Again, we simplify this expression using the order of operations. First, we perform the multiplication: -3 * 12 = -36. Then, we perform the addition: -36 + 2 = -34. Consequently, the function value f(12) is equal to -34. This means that when the input is 12, the function outputs the value -34. This process of substituting values into a function and simplifying the resulting expression is the core of function evaluation. It allows us to determine the relationship between input and output values and is essential for understanding the behavior of functions. In many mathematical contexts, you will encounter situations where you need to evaluate a function for a specific input, whether it's to find a point on a graph, solve an equation, or analyze a model. Therefore, mastering the skill of function evaluation is a crucial step in developing your mathematical proficiency.
By working through these examples, you've gained a solid understanding of how to determine whether a relation is a function and how to evaluate functions for specific input values. These are essential skills in mathematics, forming the foundation for more advanced concepts and applications.
