Function Notation: Videos & Practice Problems

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

Function Notation

Function notation is a concise way to represent functions, where instead of writing y as a function of x, we use symbols like f(x), read as "f of x." This notation expresses that f(x) is the output value corresponding to the input x. For example, if a function is defined as f(x) = 3x - 1, then f(4) means substituting 4 for x in the expression, resulting in 3(4) - 1 = 11. This shows that when the input is 4, the output is 11.

Functions can be named with different letters such as f, g, or h, but the concept remains the same: the letter represents the function, and the value inside the parentheses is the input. Sometimes, the input variable x might be omitted in notation, but the underlying idea of input-output relationships stays consistent.

When functions are given as sets of ordered pairs, the first element of each pair is the input (or x value), and the second element is the output (or y value). To evaluate a function like g(1), you locate the ordered pair where the input is 1 and identify the corresponding output. For instance, if the pair is (1, 5), then g(1) = 5.

Similarly, when a function is represented graphically, evaluating h(-2) involves finding the point on the graph where x = -2 and reading the corresponding y value. If the graph shows that at x = -2, y = 3, then h(-2) = 3.

Understanding function notation is essential for working with functions across various representations—equations, ordered pairs, or graphs. It emphasizes the relationship between inputs and outputs, allowing for clear communication and problem-solving in algebra and beyond.

Find $f(-3)$ for the following functions.
(A) $f\left(x\right)=2x^2-4x+1$

• A. $31$
• B. $29$
• C. $34$
• D. $25$

Answer: (A)

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

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $-\frac{1}{4}$
• D. $-\frac{1}{2}$

Answer: (C)

Function Notation Example 1

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 get -2y = 8 - 3x. Next, divide every term by -2 to isolate y, resulting in y = \frac{8 - 3x}{-2}. Simplifying by distributing the negative sign gives 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}. This defines the function f(x) = \frac{15 - 4x}{5}.

By solving for y and expressing it explicitly in terms of x, you create a function f(x) that maps each input x to a corresponding output y. This method ensures the equation represents a function, which is essential for understanding relationships between variables in algebra and calculus. The process highlights the importance of algebraic manipulation skills, such as adding or subtracting terms on both sides and dividing by coefficients, to isolate variables and define functions clearly.

Function Notation Example 2

The given data represents 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 are 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 represent the specific points in time for which Internet usage data is available. The range consists of the output values, which are the percentages of Internet users for each year: 43.1%, 68%, 74%, 88.5%, 90%, and 92%. This range reflects the growth in Internet adoption over the years.

Using function notation, if we define the function as f, where f(x) represents the percentage of Internet users in year x, then finding f(2015) means determining the Internet usage percentage in 2015. According to the data, f(2015) = 88.5, indicating that 88.5% of the US population used the Internet in that year.

Conversely, if we know the output value and want to find the corresponding input, such as when f(x) = 92, we look for the year when Internet usage reached 92%. The table shows this occurred in 2022, so x = 2022. This demonstrates how functions can be used to interpret real-world data by mapping inputs to outputs and vice versa.

Understanding the domain and range in this context is crucial for interpreting data trends and making predictions. The function defined by this table is a clear example of how discrete data points can represent a function, emphasizing the importance of unique input-output pairs in defining functional relationships.

Here’s what students ask on this topic:

What is function notation and how do you interpret f(x)?

Function notation is a way to represent functions using symbols like $f\left(x\right)$. Here, $f$ is the name of the function, and $x$ is the input value. The expression $f\left(x\right)$ represents the output of the function when the input is $x$. For example, if $f\left(x\right)=\; 3x\; -\; 1$, then $f\left(4\right)=\; 3\; imes\; 4\; -\; 1\; =\; 11$. This means when the input is 4, the output is 11. Function notation helps clearly show the relationship between inputs and outputs and is used to evaluate functions at specific values.

How do you evaluate a function given by an equation using function notation?

To evaluate a function given by an equation in function notation, you substitute the input value into the function's expression. For example, if the function is $f\left(x\right)=\; 3x\; -\; 1$ and you want to find $f\left(4\right)$, replace $x$ with 4: $3\; imes\; 4\; -\; 1$. Then simplify: $12\; -\; 1\; =\; 11$. So, $f\left(4\right)=\; 11$. This process applies to any input value; just replace $x$ with the given number and simplify the expression to find the output.

Can functions have different names other than f(x), and does it change how you evaluate them?

Yes, functions can have different names such as $g\left(x\right)$ or $h\left(x\right)$. The letter used is just a label and does not change how you evaluate the function. The process remains the same: substitute the input value into the function's expression or use the input to find the corresponding output from ordered pairs or a graph. For example, if $g\left(x\right)=\; 2x\; +\; 5$, then $g\left(3\right)=\; 2\; imes\; 3\; +\; 5\; =\; 11$. The function name is just a way to distinguish different functions.

How do you find the output of a function from ordered pairs using function notation?

When a function is given as ordered pairs, each pair represents an input and its corresponding output, where the first number is the input (x-value) and the second is the output (y-value). To find the output for a specific input using function notation, locate the ordered pair with that input. For example, if the function $g$ is given by the pairs $(1,\; 5),\; (2,\; 7),\; (3,\; 9)$, then $g\left(1\right)=\; 5$ because when the input is 1, the output is 5. This method helps you evaluate functions without an explicit formula.

How do you evaluate a function from a graph using function notation?

To evaluate a function from a graph using function notation, identify the input value on the x-axis, then find the point on the graph directly above or below that input. The y-coordinate of that point is the output. For example, to find $h(-2)$, locate $-2$ on the x-axis, then move vertically to the curve. The y-value at this point is the output. If the y-value is 3, then $h(-2)=\; 3$. This visual method helps understand the function's behavior and output values.