Functions And Function Notation Worksheet

Mastering Functions and Function Notation: A Comprehensive Guide with Worksheet

Understanding functions is fundamental to success in algebra and beyond. This comprehensive guide will walk you through the core concepts of functions and function notation, providing clear explanations, worked examples, and a practice worksheet to solidify your understanding. We'll cover everything from identifying functions to evaluating them using different notations, ensuring you're well-equipped to tackle even the most challenging problems.

What is a Function?

At its heart, a function is a relationship between two sets of values, often denoted as x and y, where each input (x) corresponds to exactly one output (y). Think of it like a machine: you feed it an input (x), and it produces a single, specific output (y). This "one input, one output" rule is crucial. If a single input produces multiple outputs, it's not a function.

Consider the relationship between the number of hours worked and the amount of money earned. For every hour worked (input), there's a specific amount earned (output). This is a functional relationship. However, if we considered the relationship between a person's age and their height, it wouldn't be a function because people of the same age can have different heights.

We can represent functions using various methods:

• Mappings: These visually represent the input-output relationship using arrows.
• Tables: Tables organize input and output values in a structured format.
• Graphs: Graphs provide a visual representation of the function's behavior.
• Equations: Equations define the relationship between the input and output algebraically. This is the most common and powerful representation.

Function Notation: Understanding f(x)

Function notation is a concise and powerful way to represent functions. Instead of writing "y = 2x + 1," we use function notation: f(x) = 2x + 1. This reads as "f of x equals 2x plus 1." Here, 'f' represents the function's name, and 'x' represents the input variable. f(x) represents the output value for a given input x.

The beauty of function notation lies in its clarity and efficiency. It clearly indicates the input and output, making it easier to evaluate the function for specific input values. For example, to find the output when the input is 3, we write:

f(3) = 2(3) + 1 = 7

This shows that when x = 3, the output of the function f(x) is 7.

Types of Functions

Several types of functions are frequently encountered in mathematics:

• Linear Functions: These functions have a constant rate of change and are represented by equations of the form f(x) = mx + b, where 'm' is the slope and 'b' is the y-intercept. Their graphs are straight lines.

• Quadratic Functions: These functions are represented by equations of the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. Their graphs are parabolas.

• Polynomial Functions: These functions are sums of terms of the form axⁿ, where 'n' is a non-negative integer. Linear and quadratic functions are special cases of polynomial functions.

• Rational Functions: These functions are ratios of two polynomial functions, in the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, and Q(x) ≠ 0.

• Exponential Functions: These functions have the variable in the exponent, often in the form f(x) = a^x, where 'a' is a positive constant.

• Logarithmic Functions: These are the inverse functions of exponential functions. They are represented as f(x) = logₐ(x), where 'a' is the base.

Evaluating Functions

Evaluating functions involves substituting a given value for the input variable (x) into the function's equation and simplifying to find the corresponding output value. Let's look at some examples:

Example 1:

Given g(x) = x² - 4x + 3, find g(2) and g(-1).

• g(2) = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1
• g(-1) = (-1)² - 4(-1) + 3 = 1 + 4 + 3 = 8

Example 2:

Given h(x) = √(x + 5), find h(4) and h(-1).

• h(4) = √(4 + 5) = √9 = 3
• h(-1) = √(-1 + 5) = √4 = 2

Example 3: Piecewise Functions

Piecewise functions are defined differently for different intervals of the input variable. For instance:

f(x) = {
  x + 2,  if x < 0
  x²,    if x ≥ 0
}

To evaluate, determine which part of the function applies to the given input.

• f(-2) = -2 + 2 = 0 (because -2 < 0)
• f(3) = 3² = 9 (because 3 ≥ 0)

Domain and Range

• Domain: The domain of a function is the set of all possible input values (x) for which the function is defined. For example, the domain of f(x) = √x is all non-negative real numbers (x ≥ 0) because you can't take the square root of a negative number. The domain of a rational function excludes any values of x that would make the denominator zero.

• Range: The range of a function is the set of all possible output values (y) that the function can produce. Finding the range can be more challenging and often requires examining the function's graph or understanding its behavior.

Function Composition

Function composition involves applying one function to the output of another function. If we have two functions, f(x) and g(x), the composition of f with g is denoted as (f ∘ g)(x) or f(g(x)). This means we first evaluate g(x) and then use the result as the input for f(x).

Example:

Let f(x) = x² and g(x) = x + 1. Find (f ∘ g)(x) and (g ∘ f)(x).

• (f ∘ g)(x) = f(g(x)) = f(x + 1) = (x + 1)²
• (g ∘ f)(x) = g(f(x)) = g(x²) = x² + 1

Notice that (f ∘ g)(x) and (g ∘ f)(x) are generally not equal.

Inverse Functions

An inverse function, denoted as f⁻¹(x), "undoes" the action of the original function f(x). If f(a) = b, then f⁻¹(b) = a. Not all functions have inverse functions. A function must be one-to-one (each output corresponds to only one input) to have an inverse.

Functions and Graphs

Graphs provide a visual representation of functions, allowing us to easily identify key features such as intercepts, slope, and behavior. The vertical line test can determine if a graph represents a function: if any vertical line intersects the graph more than once, it's not a function.

Worksheet: Functions and Function Notation

Now, let's test your understanding with a worksheet. Remember to show your work for each problem.

Part 1: Identifying Functions

1. Determine whether the following relations represent functions:

   a) {(1, 2), (2, 4), (3, 6), (4, 8)} b) {(1, 2), (2, 4), (3, 6), (1, 8)} c) A graph showing a circle. d) A graph showing a parabola that opens upwards.

2. Explain why a vertical line test is used to determine if a graph represents a function.

Part 2: Evaluating Functions

Given the functions below, evaluate the indicated values:

f(x) = 3x - 5

g(x) = x² + 2x - 1

h(x) = √(2x + 6)

1. f(2)
2. g(-3)
3. h(7)
4. f(0)
5. g(0)
6. h(-1)
7. f(a) (Express in terms of 'a')
8. g(a+1) (Express in terms of 'a')

Part 3: Function Composition

Given f(x) = 2x + 1 and g(x) = x - 3, find:

1. (f ∘ g)(x)
2. (g ∘ f)(x)
3. (f ∘ g)(4)
4. (g ∘ f)(0)

Part 4: Domain and Range

Determine the domain and range of the following functions:

1. f(x) = 5x + 2
2. g(x) = x²
3. h(x) = 1/(x-3)
4. i(x) = √(x-4)

Part 5: Word Problems

1. A taxi charges a flat fee of $5 plus $2 per mile. Write a function that represents the total cost (C) in terms of the number of miles (m). What is the cost of a 10-mile ride?

2. The height of a ball thrown upwards is given by the function h(t) = -16t² + 64t + 80, where h is the height in feet and t is the time in seconds. Find the height of the ball after 2 seconds.

This comprehensive guide and worksheet provide a solid foundation in understanding functions and function notation. By working through the examples and completing the worksheet, you'll develop the skills necessary to confidently tackle more advanced concepts in algebra and calculus. Remember to consult your textbook or teacher for further assistance if needed. Good luck!