This worksheet provides a comprehensive guide to evaluating functions, covering various function types and complexities. Understanding how to evaluate functions is crucial in algebra and beyond, forming the foundation for calculus, data analysis, and numerous other applications. We'll cover the basics, delve into more challenging examples, and provide answers to solidify your understanding.
What is a Function?
Before we dive into evaluating functions, let's briefly recap the definition. A function is a relationship between two sets, called the domain and the range, where each element in the domain is associated with exactly one element in the range. We often represent functions using function notation: f(x), where f is the name of the function and x represents the input (from the domain). f(x) denotes the output (from the range) corresponding to the input x.
Evaluating Functions: Basic Examples
Let's start with some simple examples to illustrate the process.
Example 1:
Given the function f(x) = 2x + 1, evaluate f(3).
Solution: To evaluate f(3), we substitute 3 for x in the function:
f(3) = 2(3) + 1 = 6 + 1 = 7
Example 2:
Given the function g(x) = x² - 4, evaluate g(-2).
Solution: Substitute -2 for x:
g(-2) = (-2)² - 4 = 4 - 4 = 0
Evaluating Functions: More Complex Examples
Now let's move on to more challenging scenarios.
Example 3:
Given the function h(x) = √(x + 5), evaluate h(4).
Solution:
h(4) = √(4 + 5) = √9 = 3
Example 4:
Given the piecewise function:
f(x) = { x² if x ≥ 0
{ -x if x < 0
Evaluate f(2) and f(-2).
Solution:
- For
f(2), since 2 ≥ 0, we use the first part of the piecewise function:f(2) = 2² = 4 - For
f(-2), since -2 < 0, we use the second part:f(-2) = -(-2) = 2
Evaluating Functions with Multiple Variables
Functions can also have multiple variables.
Example 5:
Given the function f(x, y) = x + 2y, evaluate f(3, 4).
Solution:
f(3, 4) = 3 + 2(4) = 3 + 8 = 11
Common Mistakes to Avoid
- Order of operations: Always follow the correct order of operations (PEMDAS/BODMAS) when evaluating functions, particularly when dealing with exponents, multiplication, and addition/subtraction.
- Negative numbers: Be careful when substituting negative numbers, especially when squaring or taking the square root. Remember that (-a)² = a².
- Piecewise functions: Pay close attention to the conditions defining each part of a piecewise function to ensure you're using the correct equation for the given input.
Practice Problems
- If
f(x) = 3x - 5, findf(2). - If
g(x) = x³ + 2x, findg(-1). - If
h(x) = |x - 1|, findh(5)andh(-3). - Evaluate
f(2, 1)forf(x, y) = x² - y.
Answers to Practice Problems
f(2) = 1g(-1) = -1h(5) = 4,h(-3) = 4f(2, 1) = 3
This worksheet provides a solid foundation for understanding and evaluating functions. Remember that consistent practice is key to mastering this essential mathematical skill. Further exploration of different function types, such as exponential, logarithmic, and trigonometric functions, will build upon this knowledge.
