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

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 given by the set of ordered pairs, to find g(1), you locate the pair where the input is 1 and identify the output. If the pair is (1, 5), then g(1) = 5.

When functions are graphed, the input corresponds to the x-coordinate, and the output corresponds to the y-coordinate on the graph. To find h(-2) for a function h(x) graphed as a curve, locate the point where x = -2 on the x-axis, then find the corresponding y-value on the curve. 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.

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. $\frac12$
• B. $\frac14$
• C. $-\frac14$
• D. $-\frac12$

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 fundamental in understanding relationships between variables in algebra. 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.

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. These values indicate the proportion of the population using the Internet at each corresponding year.

When defining this function as f, where f(x) represents the percentage of Internet users in year x, evaluating f(2015) means finding the output when the input is 2015. According to the data, 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 task is to determine the input year x that corresponds to this percentage. From the data, 92% Internet usage occurred in 2022, so x = 2022. This demonstrates how function notation can be used to interpret and analyze real-world data by linking inputs and outputs clearly.

Understanding how to determine whether a table defines a function, identifying the domain and range, and evaluating or solving for inputs and outputs using function notation are fundamental skills in interpreting data relationships. These concepts are essential for analyzing trends and making predictions based on tabular data.

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 $f\left(x\right)$. It means the output value of the function when the input is $x$. For example, if you have $f\left(x\right)=\; 3x\; -\; 1$, then $f\left(4\right)$ means you substitute 4 for $x$ and calculate the result. So, $f\left(4\right)=\; 3\; imes\; 4\; -\; 1\; =\; 11$. This notation helps clearly show the relationship between input and output values in a function.

How do you evaluate a function given an input value?

To evaluate a function at a specific input, you replace the variable $x$ in the function's expression with the given input value. For example, if $f\left(x\right)=\; 3x\; -\; 1$ and you want to find $f\left(4\right)$, substitute 4 for $x$: $3\; imes\; 4\; -\; 1\; =\; 12\; -\; 1\; =\; 11$. This process works the same way for any function, whether it is given as an equation, ordered pairs, or a graph.

Can functions have different names and how does that affect function notation?

Yes, functions can have different names such as $f$, $g$, or $h$. The letter used is just a label for the function and does not change how you work with it. For example, $g\left(x\right)$ and $h\left(x\right)$ are just different functions, but you evaluate them the same way by substituting the input value for $x$. The notation remains consistent regardless of the function's name.

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

When a function is given as ordered pairs, each pair represents an input and its corresponding output, usually written as $(x,\; y)$. To find the output for a specific input, look for the pair where the first value matches the input. For example, if you want to find $g(1)$ and the ordered pairs include $(1,\; 5)$, then $g(1)\; =\; 5$. This means the output when the input is 1 is 5.

How do you evaluate a function using a graph?

To evaluate a function from a graph at a given input $x$, locate the point on the x-axis that corresponds to the input value. Then, move vertically to the curve or line representing the function. The y-coordinate of this point on the graph is the output value, or $f(x)$. For example, if you want to find $h(-2)$, find $-2$ on the x-axis, then find the point on the curve above it. If the y-value there is 3, then $h(-2)\; =\; 3$.