Evaluating The Function K(x) = √(x+1) For Specific Values

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In mathematics, functions are fundamental building blocks that describe relationships between variables. Evaluating a function means determining its output value for a given input value. This article will delve into the process of evaluating the function k(x) = √(x+1) for a set of specified x values. We will explore the concept of function evaluation, the domain of the square root function, and the step-by-step calculations involved in finding the corresponding k(x) values. Understanding function evaluation is crucial for comprehending various mathematical concepts and their applications in real-world scenarios. By working through these examples, you will gain a solid grasp of how to substitute values into a function and interpret the results.

Understanding the Function k(x) = √(x+1)

The given function, k(x) = √(x+1), is a square root function. This means that for any input x, the function will output the square root of the expression (x+1). However, a critical consideration with square root functions is their domain. The domain of a function is the set of all possible input values for which the function produces a real number output. For square root functions, the expression inside the square root (the radicand) must be greater than or equal to zero. This is because the square root of a negative number is not a real number. Therefore, for k(x) = √(x+1), we must have (x+1) ≥ 0.

Solving this inequality gives us x ≥ -1. This means that the domain of the function k(x) is all real numbers greater than or equal to -1. When evaluating the function, we need to ensure that the input values fall within this domain. If an input value is less than -1, the function will not be defined for that value in the realm of real numbers. The square root function is a cornerstone of algebra and calculus, appearing in various contexts, from solving equations to modeling physical phenomena. Understanding its domain and range is essential for accurate analysis and interpretation of results. The function k(x) = √(x+1) is a simple yet powerful example of how functions can be used to represent relationships and perform calculations.

The Importance of Function Evaluation in Mathematics

Function evaluation is a cornerstone of mathematical analysis and problem-solving. It allows us to understand how a function behaves for different input values and to predict its output. In various fields, such as physics, engineering, and economics, functions are used to model real-world phenomena. Evaluating these functions at specific points helps us make predictions, optimize processes, and understand the underlying relationships. For instance, in physics, a function might describe the trajectory of a projectile, and evaluating the function at a specific time will tell us the projectile's position at that time. In economics, a function might represent the supply and demand curve for a product, and evaluating the function at a specific price will tell us the quantity supplied or demanded. Function evaluation is not just a theoretical exercise; it is a practical tool used to solve real-world problems. It provides a bridge between abstract mathematical concepts and concrete applications. Mastery of function evaluation techniques is, therefore, essential for anyone pursuing studies or careers in STEM fields. The ability to evaluate functions accurately and efficiently is a valuable skill that empowers us to analyze and interpret complex systems.

Now, let's evaluate the function k(x) = √(x+1) for the following values of x:

  • (a) x = -1
  • (b) x = -8
  • (c) x = 1
  • (d) x = 4
  • (e) x = 0

For each value, we will substitute it into the function and simplify the expression.

(a) k(-1)

To evaluate k(-1), we substitute x = -1 into the function k(x) = √(x+1):

k(-1) = √((-1) + 1) = √(0) = 0

Therefore, k(-1) = 0. This result is valid because -1 is within the domain of the function (x ≥ -1).

(b) k(-8)

Next, we evaluate k(-8) by substituting x = -8 into the function:

k(-8) = √((-8) + 1) = √(-7)

Since the radicand is -7, which is negative, the square root of -7 is not a real number. Therefore, k(-8) is undefined in the real number system. This is because -8 is outside the domain of the function (x ≥ -1).

(c) k(1)

To find k(1), we substitute x = 1 into the function:

k(1) = √(1 + 1) = √(2)

√(2) is an irrational number, approximately equal to 1.414. Therefore, k(1) = √(2). This result is valid because 1 is within the domain of the function.

(d) k(4)

We evaluate k(4) by substituting x = 4 into the function:

k(4) = √(4 + 1) = √(5)

√(5) is also an irrational number, approximately equal to 2.236. Therefore, k(4) = √(5). This result is valid because 4 is within the domain of the function.

(e) k(0)

Finally, we evaluate k(0) by substituting x = 0 into the function:

k(0) = √(0 + 1) = √(1) = 1

Therefore, k(0) = 1. This result is valid because 0 is within the domain of the function.

Here's a summary of the results of evaluating the function k(x) = √(x+1) for the given values of x:

  • (a) k(-1) = 0
  • (b) k(-8) = Undefined (not a real number)
  • (c) k(1) = √(2)
  • (d) k(4) = √(5)
  • (e) k(0) = 1

These results demonstrate how the output of the function k(x) changes as the input x varies. It also highlights the importance of considering the domain of a function when evaluating it. Values outside the domain will result in undefined outputs (in the real number system, for this square root function).

Visualizing Function Evaluation

Understanding function evaluation can be further enhanced by visualizing the function's graph. The graph of k(x) = √(x+1) is a curve that starts at the point (-1, 0) and increases as x increases. Each point on the graph represents a pair of (x, k(x)) values. When we evaluate the function for a specific value of x, we are essentially finding the y-coordinate (k(x)) of the point on the graph that corresponds to that x-coordinate. For instance, the point (0, 1) lies on the graph of k(x) = √(x+1), which visually represents the result k(0) = 1. Similarly, the point (1, √(2)) lies on the graph, representing the result k(1) = √(2). Visualizing the graph helps to solidify the concept of function evaluation and provides a geometric interpretation of the relationship between input and output values. The graph also clearly shows the domain of the function, as there are no points on the graph for x values less than -1.

In this article, we evaluated the function k(x) = √(x+1) for various values of x. We emphasized the importance of understanding the domain of a function, especially for square root functions, where the radicand must be non-negative. We demonstrated the step-by-step process of substituting values into the function and simplifying the resulting expressions. We found that some values, like x = -8, resulted in undefined outputs because they fell outside the function's domain. Other values, like x = -1, 0, 1, and 4, produced real number outputs. Function evaluation is a fundamental skill in mathematics, and this exercise provides a solid foundation for understanding more complex functions and mathematical concepts. By practicing function evaluation with different types of functions, you can strengthen your mathematical skills and prepare for advanced topics in algebra, calculus, and other areas of mathematics. The ability to accurately evaluate functions is a valuable asset in both academic and professional settings, enabling you to solve problems and make informed decisions based on mathematical models.