PC - Functions Lesson

Functions

A function is a rule (correspondence) that assigns to each input a unique output. Functions can be described in many ways such as by mapping/arrow diagrams, graphs, tables and even by words. In this course we mainly consider functions for which input (x) and output (y) values are sets of real numbers. It may be helpful to think of a function's input and output as an ordered pair. The input variable x is the independent variable and the output variable y is the dependent variable. View the video introducing functions below.

A relation is any set of ordered pairs such as A= { (1,2), (2,3), (3,4), (2,5)}. Notice that this relation is not a function since there is not a unique output for an input of 2.

Notation and Evaluation

In order to make it clear exactly what rule associates an input value with an output value, a special mathematical notation is used. If we give the rule a name such as f, then the mapping (arrow) diagram below conveys an input of x is acted upon to produce an output of f(x), which is function notation for what was formerly denoted as y. Recall that f(x) is read "f of x".

 $x\xrightarrow{\text{f}} f\left(x\right)$

Functions are typically named with the letters f, g, and h, although any nomenclature is acceptable, even something like the following:

$x\xrightarrow{\text{calculusisawesome}} calculusisawesome\left(x\right)$

In this case x is still the input/independent variable and calculusisawesome(x) is the output/dependent variable.

Evaluating a function means finding the output value for a given input value. If x = -1 and the function rule is h(x) = x²- 3x, then the function value when x = -1 is:

h(-1) = (-1)² - 3(-1) = 1 + 3 = 4

This function's input and output expressed as an ordered pair is (x, h(x)) = (-1, 4). 

Domain and Range

The domain of a function f is the set of all possible x's (inputs). The range of a function f is the set of all possible output values f(x). In function language each input x is often described as the preimage and each associated output f(x) is referred to as the image. View the< Determining Domain and Range video below

One-to-One

A function f that never takes on the same value twice is called a one-to-one function. In terms of function notation, this means that $f\left(x_1\right)\ne f\left(x_2\right)$ whenever $x_1\ne x_2$. Thinking of this in a real-life context, different combination locks (x) must have different combinations(y).

Graphically, you can think of a one-to-one function as different x values must not have the same y values. The first graphed function below is one-to-one but the second graphed function is not one-to-one.

Just as the vertical-line test is a quick way to visually determine if a graph represents a function, the horizontal line test is helpful in determining if a function is one-to-one.

• Any horizontal line drawn on the first graph above intersects the curve only once, which means the function is one-to-one, i.e., whenever $x_1\ne x_2$ then $f\left(x_1\right)\ne f\left(x_2\right)$.
• Any horizontal line drawn on the second curve above intersects the curve more than once, which shows the function is not one-to-one, i.e., whenever $x_1\ne x_2$ then $f\left(x_1\right)=f\left(x_2\right)$.

Combining and Composing

Two functions, f and g can be combined to create new functions f + g, f - g, fg, and f/g in the same way we add, subtract, multiply, and divide real numbers. Click HERE to view the example below. Links to an external site.

Given the functions f and g, the composition function f ο g (read f of g) is defined to be $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$. The domain of f ο g is the set of all x in the domain of g such that g(x) is in the domain of f. To find the domain of a composite function, two steps are needed:

1. Find the domain of the "inside" (input) function. If there are any restrictions on the domain, keep them.
2. Create the composite function and find the domain of this new function. If there are restrictions on this domain, add them to the restrictions from Step 1. If there is an overlap, use the more restrictive domain (or the intersection of the domains). Note: The composite may result in a domain unrelated to the domains of the original functions.

For example, if f(x) = x² + 2 and g(x) = √(3 - x), then (f ο g)(x) = f(g(x)) = (√(3 - x))² + 2 = 5 - x. The domain of g(x) is x < 3 and the domain for f(g(x)) is all real numbers. Although the resulting function is linear and has no domain restrictions, the domain for the composite function is also   x   <   3 because the initial function g(x) = √(3 - x) has a domain restriction.

Even or Odd

If a function f satisfies f(-x) = f(x) for every x in the function's domain, then f is an even function. Geometrically, the graph of an even function is symmetric about the y-axis.

If a function f satisfies f(-x) = - f(x) for every x in the function's domain, then f is an odd function. Geometrically, the graph of an odd function is symmetric about the origin.

The terms even and odd relate to the power of x. If $f\left(x\right)=x^2\:or\:f\left(x\right)=x^4$  or x raised to any even power, then  .

If $f\left(x\right)=x^3$  or $x^5$ or x raised to any odd power, then $f\left(-x\right)=\left(-x\right)^5=-x^5=-f\left(x\right)$.

Note that some functions such as $g\left(x\right)=x^3+x^2$ are neither even nor odd since $g\left(-x\right)=\left(-x\right)^3+\left(-x\right)^2=-x^3+x^2$. This result isn't the original function or its opposite. Special note; remember that while zero is neither positive or negative, it is considered an even number. Therefore, a constant is considered "even." For example, y = x² + 3 is an even function, as 3 is actually 3x ⁰. This can also be seen in the graph, with y-axis symmetry.

Functions Practice

Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the function-related content.

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