Evaluating Functions: A Comprehensive Worksheet with Answers
This worksheet provides a thorough exploration of evaluating functions, a fundamental concept in algebra. We'll cover various function types and techniques to help you master this skill. Whether you're a student looking to improve your algebra skills or a teacher seeking resources for your classroom, this comprehensive guide will serve as a valuable tool.
What is a Function?
Before diving into evaluating functions, let's define what a function is. A function is a relationship between two sets, typically denoted as x (the input or domain) and y (the output or range), where each input value corresponds to exactly one output value. Functions are often expressed using function notation, such as f(x), g(x), or h(x), where the letter represents the function's name and the variable x represents the input.
Evaluating Functions: The Basics
Evaluating a function involves substituting a given value for the input variable (x) and then simplifying the expression to find the corresponding output value (y). Let's look at some examples:
Example 1:
If f(x) = 2x + 3, find f(4).
To find f(4), we substitute 4 for x in the function:
f(4) = 2(4) + 3 = 8 + 3 = 11
Therefore, f(4) = 11.
Example 2:
If g(x) = x² - 5x + 6, find g(-2).
Substituting -2 for x:
g(-2) = (-2)² - 5(-2) + 6 = 4 + 10 + 6 = 20
Therefore, g(-2) = 20.
Different Types of Functions and Evaluation Techniques
While the examples above demonstrate basic function evaluation, various types of functions exist, each with its evaluation nuances.
Evaluating Piecewise Functions
Piecewise functions are defined by multiple sub-functions, each applicable over a specific interval of the domain. Evaluating piecewise functions requires determining which sub-function applies based on the input value.
Example 3:
Consider the piecewise function:
f(x) = { x + 1, if x < 0
{ x² , if x ≥ 0
Find f(-2) and f(3).
For f(-2), since -2 < 0, we use the first sub-function: f(-2) = -2 + 1 = -1.
For f(3), since 3 ≥ 0, we use the second sub-function: f(3) = 3² = 9.
Evaluating Functions with Multiple Variables
Functions can have more than one input variable. Evaluating these functions requires substituting values for each variable.
Example 4:
If h(x, y) = 3x + 2y, find h(2, 5).
Substituting x = 2 and y = 5:
h(2, 5) = 3(2) + 2(5) = 6 + 10 = 16
Therefore, h(2, 5) = 16.
Evaluating Rational Functions
Rational functions are fractions where both the numerator and denominator are polynomials. Care must be taken to avoid division by zero.
Example 5:
If r(x) = (x + 1) / (x - 2), find r(3).
Substituting x = 3:
r(3) = (3 + 1) / (3 - 2) = 4 / 1 = 4
However, r(2) is undefined because it would result in division by zero.
Frequently Asked Questions (FAQs)
This section addresses common questions surrounding evaluating functions:
Q: What if the function involves absolute values?
A: When evaluating functions with absolute values, remember that the absolute value of a number is its distance from zero, always resulting in a non-negative value. For instance, |3| = 3 and |-3| = 3. Simply substitute the input value and evaluate the expression according to the rules of absolute value.
Q: How do I handle functions with radicals (square roots)?
A: When evaluating functions containing radicals, ensure that the expression inside the radical is non-negative. If the expression inside the radical becomes negative after substitution, the function is undefined for that particular input value.
Q: What if the function is a composition of functions?
A: For composite functions, evaluate the inner function first, then substitute the result into the outer function.
This worksheet aims to provide a strong foundation in evaluating functions. Remember, practice is key to mastering this skill. Work through various examples and types of functions to solidify your understanding. Further resources can be found online and in textbooks dedicated to algebra and precalculus.
