Functions

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SAT Functions Tutorial

In this tutorial, we will cover all the essential topics related to functions that appear in the SAT exam. The SAT often tests your understanding of functions, their properties, and how to manipulate them.

Introduction to Functions

In mathematics, functions are fundamental concepts used to describe relationships between different quantities. They play a crucial role in various fields, from pure mathematics to real-world applications in physics, engineering, economics, and more. Understanding functions is essential for success in standardized tests like the SAT and is a foundational skill for higher-level mathematics courses.

What is a Function?

A function is a special kind of relation that associates each element from one set, called the domain, to a unique element in another set, called the codomain or range. The key property of functions is that each input value in the domain corresponds to exactly one output value in the codomain.

Mathematically, a function can be represented by the notation:

f: X → Y

where f is the name of the function, X is the domain, and Y is the codomain. For a given input value x in the domain, the corresponding output value is denoted as f(x).

Function Notation

To define a function explicitly, we often use function notation, where we represent the function as a rule or formula. In this notation, we use the function name (such as f, g, h, etc.) followed by an open parenthesis, the input variable (usually x), and then a closing parenthesis. The output of the function is the value obtained after substituting the input variable into the expression.

For example, suppose we have a function f(x) = 2x + 3. To evaluate f at a specific value of x, we simply replace x with that value in the expression. If we want to find f at x = 5, we have:

f(5) = 2(5) + 3
f(5) = 13

Domain and Range

The domain of a function is the set of all possible input values for which the function is defined. It represents the values for which the function makes sense and produces valid outputs. The domain can be any subset of the real numbers, or it can be restricted to certain values based on the context of the problem.

The range of a function, on the other hand, is the set of all possible output values that the function can produce. It represents the values that the function can take on based on its domain and rule.

Evaluating Functions

Evaluating a function simply means finding the output value (f(x)) for a given input value (x). To do this, you substitute the input value into the function expression and calculate the result.

For instance, let's consider the function f(x) = 2x^2 - 3x + 1. To evaluate f at x = 2, we have:

f(2) = 2(2)^2 - 3(2) + 1
f(2) = 2(4) - 6 + 1
f(2) = 8 - 6 + 1
f(2) = 3

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Graphing a function

Graphing a function is a visual representation of how the function behaves across a range of input values. It allows us to observe the shape, behavior, and key characteristics of the function. When plotting a graph, we follow certain steps to ensure an accurate representation. Here's how you can plot the graph of a function:

Step 1: Determine the Domain

First, identify the domain of the function, which represents the set of valid input values. Some functions may have restrictions on the domain due to division by zero, square roots of negative numbers (for real numbers), or other mathematical constraints. The domain is crucial because it defines the range of x values for which the graph is meaningful.

Step 2: Choose Representative Points

Select several representative points within the domain of the function. Typically, three to five points are sufficient for most functions. You can choose values of x from the domain and evaluate the function to obtain the corresponding y values.

Step 3: Plot the Points

Using the x and y values obtained from Step 2, plot the points on a Cartesian coordinate system. Place each point at the intersection of its x and y values. It's helpful to label the points with their coordinates to avoid confusion.

Step 4: Draw the Graph

Connect the plotted points with a smooth curve to create the graph of the function. The curve should reflect the behavior of the function between the selected points. For linear functions, the graph will be a straight line. For quadratic functions, it will be a parabola, and for other functions, it may be more complex.

Example

Let's use a linear function as an example to follow the steps for graphing:

Consider the function f(x) = 2x + 1.

Step 1: Determine the Domain

The domain of the function f(x) is all real numbers because there are no constraints on the input x.

Step 2: Choose Representative Points

Let's choose three representative points within a reasonable range, such as -2, 0, and 3. We will find the corresponding y values by evaluating the function:

• For x = -2: f(-2) = 2(-2) + 1 = -4 + 1 = -3
• For x = 0: f(0) = 2(0) + 1 = 0 + 1 = 1
• For x = 3: f(3) = 2(3) + 1 = 6 + 1 = 7

Step 3: Plot the Points

Now, we plot the points (x, f(x)):

• Point 1: (-2, -3)
• Point 2: (0, 1)
• Point 3: (3, 7)

Step 4: Draw the Graph

Next, we connect the plotted points with a straight line since our function is linear.

Composite Functions

Composite functions arise when we combine two functions to form a new function. The output of one function becomes the input for the other. This concept allows us to model more complex relationships between quantities.

Suppose we have two functions f(x) and g(x). The composite function h(x) can be defined as:

h(x) = f(g(x))

In this case, the output of g(x) becomes the input of f(x). To evaluate h(x) at a particular value of x, we first find g(x) and then use the result as the input for f(x).

Let's consider two functions f(x) = 2x + 3 and g(x) = x^2. We will find the composite function h(x) = f(g(x)) at x = 4.

Step 1: Find g(x)

Given g(x) = x^2, let's find g(4):

g(4) = 4^2
g(4) = 16

Step 2: Find f(g(x))

Now that we know g(4) = 16, we can find f(g(x)) by using x = 16 as the input for f(x) = 2x + 3:

f(g(x)) = f(16) = 2(16) + 3
f(g(x)) = 32 + 3
f(g(x)) = 35

So, the composite function h(x) = f(g(x)) is equal to 35. This means that for any value of x you choose, if you first evaluate g(x) and then use that result as the input for f(x), the output will always be 35.

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