4. Graphs and Functions

Function Notation

4. Graphs and Functions

Function Notation: Videos & Practice Problems

Topic summary

Function notation, such as $f (x)$ , represents the output value of a function for a given input $x$ . Evaluating functions involves substituting the input into the expression, for example, $f (4) = 3×4−1 = 11$ . Functions can be named differently (e.g., $g (x)$ , $h (x)$ ) but follow the same principles. Inputs correspond to x-values, and outputs to y-values, which can be found from ordered pairs or graphs, reinforcing understanding of polynomial terms and their evaluation.

concept

Function Notation

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Function Notation Video Summary

Functions can be expressed using a special notation called function notation, which provides a clear way to represent the relationship between inputs and outputs. Instead of writing an equation as y = 3x - 1, we use the notation f(x) = 3x - 1, where f is the name of the function and x is the input variable. This notation is read as "f of x" and represents the output value corresponding to the input x.

To evaluate a function at a specific input, such as f(4), you substitute the input value into the function's expression. For example, given f(x) = 3x - 1, substituting x = 4 yields:

\[f(4) = 3 \times 4 - 1 = 12 - 1 = 11\]

This means when the input is 4, the output of the function is 11. Function notation is versatile and can use different letters to name functions, such as g(x) or h(x), but the process of substitution and evaluation remains the same.

Functions can also be represented as sets of ordered pairs, where each pair consists of an input (x-value) and its corresponding output (y-value). For example, if a function g is defined by the ordered pairs {(1, 5), (2, 7), (3, 9)}, then g(1) equals 5 because the input 1 corresponds to the output 5.

When functions are graphed, the input values correspond to points on the x-axis, and the output values correspond to points on the y-axis. To find the output for a given input, locate the input value on the x-axis, then find the point on the curve directly above or below it. For instance, if the function h(x) is graphed and you want to find h(-2), you find -2 on the x-axis and then identify the y-value of the point on the curve at that x-coordinate. If the y-value is 3, then h(-2) = 3.

Understanding function notation is essential for working with functions in algebra and calculus. It allows you to clearly express and evaluate functions, whether given by equations, ordered pairs, or graphs. Remember, the key idea is that the function takes an input x and produces an output, often denoted as f(x), g(x), or h(x), depending on the function's name.

Problem

Find $f of negative 3$ for the following functions.
(A) $f (x) = 2 x^{2} - 4 x + 1$

$31$

$29$

$34$

$25$

Problem

Find $f of negative 3$ for the following functions.
(B) $f (x) = \frac{x + 4}{x - 1}$

$\frac{1}{2}$

$\frac{1}{4}$

$-\frac14$

$-\frac12$

example

Function Notation Example 1

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Function Notation Example 1 Video Summary

To define a function f(x) that expresses y as a function of x, the key step is to solve the given equation for y. This process involves isolating y on one side of the equation, which then allows you to rewrite the equation in the form y = f(x). For example, consider the equation -3x - 2y = 8. First, subtract -3x from both sides to eliminate the x term on the left, resulting in -2y = 8 - 3x. Next, divide every term by -2 to isolate y, giving y = \frac{8 - 3x}{-2}. Simplifying by distributing the negative sign in the numerator, this becomes y = \frac{3x - 8}{2}. This expression defines the function f(x) = \frac{3x - 8}{2}.

Similarly, for the equation 4x + 5y = 15, isolate y by subtracting 4x from both sides, yielding 5y = 15 - 4x. Dividing both sides by 5 gives y = \frac{15 - 4x}{5}, which defines the function f(x) = \frac{15 - 4x}{5}. This method of solving for y and then expressing it as a function of x is fundamental in algebra for converting implicit equations into explicit function notation.

Understanding how to manipulate equations to isolate variables is essential for defining functions and analyzing relationships between variables. This approach not only clarifies the dependency of y on x but also facilitates further operations such as graphing, evaluating, and applying functions in various mathematical contexts.

example

Function Notation Example 2

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Function Notation Example 2 Video Summary

The given data presents the percentage of the US population using the Internet over various years, which can be analyzed as a function. In this context, the years serve as the input values (domain), and the corresponding Internet usage percentages represent the output values (range). Since each year corresponds to exactly one percentage value, this relationship satisfies the definition of a function, where every input has a unique output.

To identify the domain, list all the input years: 2000, 2005, 2010, 2015, 2020, and 2022. These discrete values represent the set of all possible inputs for the function. The range consists of the output percentages: 43.1, 68, 74, 88.5, 90, and 92, which are the Internet usage rates corresponding to each year.

Using function notation, if we define the function as f, then f(x) represents the Internet usage percentage for the year x. For example, to find f(2015), locate the input 2015 in the domain and observe the corresponding output, which is 88.5. Thus, f(2015) = 88.5, indicating that 88.5% of the US population used the Internet in 2015.

Conversely, if given an output value such as f(x) = 92, the goal is to determine the input x that produces this output. By examining the data, 92 corresponds to the year 2022, so x = 2022. This means that in 2022, 92% of the US population used the Internet.

This example illustrates how to interpret tabular data as a function, identify its domain and range, and apply function notation to find specific input-output relationships. Understanding these concepts is fundamental in analyzing real-world data sets and modeling relationships between variables.

Here’s what students ask on this topic:

Function notation is a way to represent functions using symbols like $f (x)$ , which means "f of x." It shows the output value of a function for a given input $x$ . Instead of writing $y = 3x - 1$ , we write $f (x) = 3x - 1$ . Here, $f (x)$ represents the output when you input $x$ . To evaluate the function at a specific value, say 4, you substitute 4 for $x$ and simplify: $f (4) = 3 imes 4 - 1 = 11$ . This notation helps clearly distinguish the function's name and its input, making it easier to work with functions in algebra.

When a function is given as a set of ordered pairs, each pair represents an input and its corresponding output, usually written as $(x, y)$ . To evaluate the function at a specific input, say $x = 1$ , you look for the ordered pair where the first value is 1. For example, if the set includes $(1, 5)$ , then the output for input 1 is 5. So, $f(1) = 5$ . This method is straightforward because you directly find the output value associated with the input from the pairs, reinforcing the idea that functions map inputs to outputs.

To find the value of a function from its graph at a specific input $x$ , first locate the point on the x-axis corresponding to that input. Then, move vertically to the curve of the function. The y-coordinate of the point on the curve above (or below) the input is the output value of the function. For example, if you want to find $h(-2)$ , find $x = -2$ on the x-axis, then look at the height of the curve at that point. If the curve is at $y = 3$ , then $h(-2) = 3$ . This visual method helps understand how functions assign outputs to inputs graphically.

Yes, functions can have different names such as $f(x)$ , $g(x)$ , or $h(x)$ . The letter used is simply the function's name and does not change how the function works. The notation remains the same: the letter represents the function, and the value inside the parentheses is the input. For example, $g(2)$ means the output of function $g$ when the input is 2. Regardless of the letter, the process of evaluating the function by substituting the input value stays consistent. This flexibility allows multiple functions to be discussed clearly and separately.

In function notation, the input is the value you substitute into the function, usually represented by $x$ . The output is the result you get after evaluating the function at that input, often represented by $f(x)$ . For example, in $f(x) = 3x - 1$ , if the input is 4, then the output is $f(4) = 3 imes 4 - 1 = 11$ . Inputs correspond to the x-values, and outputs correspond to the y-values or function values. Understanding this distinction is crucial for working with functions effectively.