Definition of a function

MAT1510 - Precalculus Mathematics A · Functions and Graphs

Definition of a Function

A function is a special relationship between two sets of numbers or objects. In simple terms, a function takes an input, performs a specific operation on it, and produces exactly one output. This relationship can be represented in various ways, such as equations, graphs, or tables.

Notation of Functions

Functions are commonly denoted by letters such as f, g, or h. The notation f(x) represents the output of the function f when the input is x. For example, if f(x) = 2x + 3, then:

f(1) = 2(1) + 3 = 5

In this case, when the input is 1, the output is 5.

Remember: A function must produce one and only one output for each input.

Domain and Range

The domain of a function is the complete set of possible inputs (x-values) that the function can accept. The range is the complete set of possible outputs (y-values) that the function can produce.

Example of Domain and Range

Consider the function f(x) = x^2. The domain of this function is all real numbers because you can substitute any real number for x. The range, however, is all non-negative real numbers because squaring any real number will not produce a negative result.

Tip: To find the domain and range, consider the operations involved in the function. Look for restrictions such as division by zero or square roots of negative numbers.

Types of Functions

Functions can be classified into different types based on their characteristics. Some common types include:

Linear Functions: Functions of the form f(x) = mx + b, where m and b are constants. The graph is a straight line.
Quadratic Functions: Functions of the form f(x) = ax^2 + bx + c, where a, b, and c are constants. The graph is a parabola.
Cubic Functions: Functions of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. The graph has an S-shape.

Example of a Linear Function

Let’s consider the linear function f(x) = 2x + 1. To find some points on the graph, we can calculate f(x) for several values of x:

f(-1) = 2(-1) + 1 = -1
f(0) = 2(0) + 1 = 1
f(1) = 2(1) + 1 = 3

The points (-1, -1), (0, 1), and (1, 3) can be plotted on a graph to show the straight line.

Example of a Quadratic Function

Now consider the quadratic function f(x) = x^2 - 4. We can find some points similarly:

f(-2) = (-2)^2 - 4 = 0
f(0) = (0)^2 - 4 = -4
f(2) = (2)^2 - 4 = 0

The points (-2, 0), (0, -4), and (2, 0) can be plotted to show the parabola.

Watch out: Remember that quadratic functions have a minimum or maximum point, known as the vertex. This is important for understanding their shape.

Vertical Line Test

To determine if a graph represents a function, you can use the vertical line test. If a vertical line intersects the graph at more than one point, then the graph does not represent a function. Conversely, if it intersects at most one point, it is a function.

Example of the Vertical Line Test

Consider the graph of the circle defined by the equation x^2 + y^2 = 4. If you draw a vertical line at x = 1, it will intersect the circle at two points. Therefore, this graph does not represent a function.

For x = 1, y can be:
y = √(4 - 1^2) = √3 (positive point)
y = -√(4 - 1^2) = -√3 (negative point)

Function Composition

Function composition occurs when you combine two functions. If you have two functions, f(x) and g(x), the composition of f and g, denoted as (f ∘ g)(x), means you apply g first and then apply f to the result of g.

Example of Function Composition

Let f(x) = 2x and g(x) = x + 3. The composition (f ∘ g)(x) is calculated as follows:

(f ∘ g)(x) = f(g(x))
g(x) = x + 3
f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6

Thus, (f ∘ g)(x) = 2x + 6.

Remember: When composing functions, always apply the inner function first.

Inverse Functions

An inverse function reverses the effect of the original function. If f(x) is a function, the inverse function is denoted as f^-1(x). To find the inverse, you swap the input and output and solve for the new output.

Example of Finding an Inverse Function

Consider the function f(x) = 3x - 5. To find the inverse:

Replace f(x) with y: y = 3x - 5.
Swap x and y: x = 3y - 5.
Solve for y:

x + 5 = 3y
y = (x + 5)/3

Thus, the inverse function is f^-1(x) = (x + 5)/3.

Tip: To verify if two functions are inverses, check if (f ∘ f^-1)(x) = x and (f^-1 ∘ f)(x) = x.

Summary

A function is a relationship that assigns exactly one output for each input.
The domain is the set of all possible inputs, and the range is the set of all possible outputs.
Common types of functions include linear, quadratic, and cubic functions.
The vertical line test helps determine if a graph represents a function.
Function composition combines two functions, while inverse functions reverse the effect of the original function.

Check your understanding

What is the definition of a function?
How do you determine the domain and range of a function?
What is the vertical line test and how is it used?
How do you find the inverse of a function?