Evaluating Functions Step-by-Step Solutions And Examples

July 8, 2025 by Scholario Team 57 views

Evaluating Functions A Comprehensive Guide with Solutions

In mathematics, evaluating functions is a fundamental skill. It involves substituting a given value into a function and simplifying the expression to find the output. This comprehensive guide will walk you through the process of evaluating various types of functions with detailed solutions and explanations. Whether you are a student learning the basics or someone looking to refresh your understanding, this article will provide you with the necessary tools and knowledge. We will explore different types of functions, including piecewise functions, rational functions, absolute value functions, and composite functions. Each section will include step-by-step solutions and explanations to help you grasp the concepts thoroughly. Let's dive into the world of function evaluation and enhance your mathematical skills!

1. Evaluating a Piecewise Function

Piecewise functions, as the name suggests, are functions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Evaluating these functions requires careful attention to the specified domain intervals. In our first example, we're tasked with evaluating a piecewise function, w(n), which is defined differently for n = 1 and n = -|5x - 7|. This type of function is essential in various mathematical models where different conditions lead to different outcomes. Understanding how to evaluate piecewise functions is crucial for solving problems in calculus, real analysis, and computer science. The key to mastering these functions lies in correctly identifying which sub-function applies to the given input value. For instance, if we are asked to find w(1), we directly use the first part of the definition because n is indeed equal to 1. However, if we need to find w(n) for another value, such as w(2), we would need to consider the second part of the definition, which involves an absolute value expression. This step-by-step evaluation process ensures that we apply the correct rule and obtain the accurate output for any given input. The ability to work with piecewise functions opens doors to more complex mathematical concepts and practical applications, making it a valuable skill for any math enthusiast or professional.

Solution:

The given piecewise function is:

w(n) = 
  n = 1
  n = -|5x - 7|

For n = 1, w(n) = 1.
For n = -|5x - 7|, we need to substitute different values of x to find the corresponding value of n. Let's consider a scenario where we want to find w(n) when x = 2:
- Substitute x = 2 into -|5x - 7|:
```
n = -|5(2) - 7|
n = -|10 - 7|
n = -|3|
n = -3
```
- So, when x = 2, n = -3, and w(n) = -3.

This example illustrates how to evaluate a piecewise function by considering the conditions for each part of the function. The process involves substituting the given value into the appropriate sub-function and simplifying the expression. Each part of the piecewise function has its own domain, and it is crucial to identify which domain the input value belongs to before evaluating the function.

2. Evaluating a Rational Function

Moving on, let’s consider the scenario of evaluating rational functions. These are functions expressed as the quotient of two polynomials. Rational functions often require careful consideration of the denominator to avoid division by zero, which is undefined in mathematics. The function we're examining here is f(x) = (5x - 7) / (3x - 2). To evaluate this function at x = -3, we need to substitute -3 for x in the expression and simplify. Understanding how to work with rational functions is paramount in algebra and calculus. These functions frequently appear in real-world applications, such as modeling rates of change and describing physical phenomena. The process of evaluation involves not only substitution but also simplification of the resulting expression, often requiring knowledge of algebraic manipulations. It's also crucial to check for any restrictions on the domain, such as values of x that would make the denominator zero. Mastering the evaluation of rational functions provides a solid foundation for tackling more advanced mathematical problems and understanding their applications in various fields.

Solution:

The given rational function is:

f(x) = (5x - 7) / (3x - 2)

To find f(-3), substitute x = -3:

f(-3) = (5(-3) - 7) / (3(-3) - 2)
f(-3) = (-15 - 7) / (-9 - 2)
f(-3) = (-22) / (-11)
f(-3) = 2

Therefore, f(-3) = 2.

3. Evaluating an Absolute Value Function

Absolute value functions are another important category in mathematics. Evaluating them involves understanding the concept of absolute value, which represents the distance of a number from zero, regardless of its sign. Our function here is f(x) = |x| - 3, and we're asked to find f(9.3). The absolute value function is denoted by vertical bars | |, and it transforms any negative number into its positive counterpart while leaving positive numbers unchanged. This property is fundamental in many mathematical contexts, including geometry and calculus. When evaluating absolute value functions, the key is to first determine the value inside the absolute value bars and then apply the absolute value transformation. In our case, we need to find the absolute value of 9.3, which is simply 9.3 since it’s already positive. Then, we subtract 3 from this value to get the final result. The ability to work with absolute value functions is crucial for solving equations and inequalities involving absolute values and for understanding more complex functions that incorporate absolute value expressions. These functions appear in various applications, such as modeling distances, errors, and tolerances, making them a valuable tool in both theoretical and practical mathematics.

Solution:

The given absolute value function is:

f(x) = |x| - 3

To find f(9.3), substitute x = 9.3:

f(9.3) = |9.3| - 3
f(9.3) = 9.3 - 3
f(9.3) = 6.3

Thus, f(9.3) = 6.3.

4. Evaluating a Function with a Composite Argument

Now, let's delve into evaluating functions with composite arguments. This involves substituting an expression, rather than a simple numerical value, into the function. Composite arguments can make the evaluation process slightly more complex, but it's a crucial skill for advanced mathematical concepts like function composition. In this instance, we have the function f(x) = 4x + 5, and we need to find f(2x + 3). This means we're replacing the x in the function's expression with the entire expression 2x + 3. The key to solving these problems is to perform the substitution carefully and then simplify the resulting expression using algebraic techniques. This type of evaluation is fundamental in calculus when dealing with chain rules and transformations of functions. It also appears in various applications, such as computer graphics and signal processing. By mastering the evaluation of functions with composite arguments, you'll enhance your ability to manipulate and understand more complex mathematical models and equations.

Solution:

The given function is:

f(x) = 4x + 5

To find f(2x + 3), substitute 2x + 3 for x:

f(2x + 3) = 4(2x + 3) + 5
f(2x + 3) = 8x + 12 + 5
f(2x + 3) = 8x + 17

So, f(2x + 3) = 8x + 17.

5. Evaluating a Simple Function

Lastly, we will practice evaluating a simple function. Simple functions might seem straightforward, but they form the building blocks for more complex mathematical models. In this final example, we have the function g(p) = 4x with the condition x = 3. This example highlights the importance of paying attention to the variables used in the function definition. Even though the function is defined in terms of p, the expression involves x, and we are given a specific value for x. The process of evaluating such a function involves substituting the given value of x into the expression. It is crucial to understand that the variable p serves as the input to the function g, but the actual computation depends on the value of x. This type of evaluation is common in various mathematical contexts and helps in understanding the relationship between variables in a function. Mastering the evaluation of simple functions lays the groundwork for handling more intricate functions and their applications in diverse fields such as physics, engineering, and economics.

Solution:

The given function is:

g(p) = 4x

with x = 3. To find g(p), substitute x = 3:

g(p) = 4(3)
g(p) = 12

Thus, g(p) = 12.

Conclusion

In conclusion, this article has provided a comprehensive guide to evaluating functions, covering various types such as piecewise, rational, absolute value, and composite functions. Evaluating functions is a fundamental skill in mathematics, and mastering it is crucial for success in algebra, calculus, and beyond. We have demonstrated step-by-step solutions for each type of function, emphasizing the importance of understanding the function's definition and the specific input value or expression. By carefully substituting and simplifying, you can accurately evaluate any function. Remember to pay close attention to the domain restrictions, especially for rational functions, and the properties of absolute value. With practice, you'll become proficient in evaluating functions and applying this skill to solve a wide range of mathematical problems. Keep exploring different types of functions and their applications to deepen your understanding and enhance your problem-solving abilities.