Functions can be represented in several ways, each providing a unique perspective on the relationship between inputs and outputs. Here are the primary methods:
Function Representations Practice Problems & Worksheets
Master algebraic, graphical, tabular and verbal representations of functions with step-by-step practice problems. Build confidence with interactive exercises.
📚Practice Converting Between Function Representations
- Convert algebraic equations like f(x) = 2x + 3 into tables and graphs
- Create coordinate plane graphs from function equations with proper scaling
- Write verbal descriptions of function relationships using clear mathematical language
- Build tables of values by substituting x-values into algebraic representations
- Identify increasing and decreasing functions from graphs and equations
- Switch between Y= and f(x) notation confidently in all problems
Understanding Representations of Functions
Complete explanation with examples
Ways to represent a function
Algebraic Representation
Representation using an equation of and , such as , showing how the output depends on the input.
Graphical representation
A visual representation on a coordinate plane, like using a graph, plotting on the and axis, where the function's behavior and trends (e.g., linear, quadratic) can be observed.
Tabular representation
A table of values that pairs inputs () with corresponding outputs () for a quick reference of specific points.
Verbal representation
A written explanation describing the relationship between variables, such as “The output is twice the input plus three.” Expressing the relationship between and using words.
Function notation
Practice Representations of Functions
Test your knowledge with 12 quizzes
Determine whether the following table represents a linear function
Examples with solutions for Representations of Functions
Step-by-step solutions included
Exercise #1
Is the given graph a function?
Step-by-Step Solution
To determine if the given graph represents a function, we use the vertical line test: if any vertical line intersects the graph at more than one point, the graph is not a function.
Let's apply this test to the graph:
- Examine different sections of the graph by drawing imaginary vertical lines.
- Look for intersections where more than one point exists on the vertical line.
Upon examining the graph, we observe that there are several vertical lines that intersect the graph at multiple points, particularly in areas with loops or overlapping curves. This indicates that at those -values, there are multiple -values corresponding to them.
Since there exist such vertical lines, according to the vertical line test, the graph does not represent a function.
Thus, the solution to this problem is that the given graph is not a function.
Answer:
No
Video Solution
Exercise #2
Is the given graph a function?
Step-by-Step Solution
It is important to remember that a function is an equation that assigns to each element in domain X one and only one element in range Y
Let's note that in the graph:
In other words, there are two values for the same number.
Therefore, the graph is not a function.
Answer:
No
Video Solution
Exercise #3
Determine whether the following table represents a function
Step-by-Step Solution
It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.
In the table, we can observe that there is a constant change in X values, meaning an increase of 1, and a constant change in Y values, meaning an increase of 3
Therefore, according to the rule, the table describes a function.
Answer:
Yes
Video Solution
Exercise #4
Determine whether the data in the following table represent a constant function
Step-by-Step Solution
It is important to remember that a constant function describes a situation where as the X value increases, the function value (Y) remains constant.
In the table, we can observe that there is a constant change in X values, meaning an increase of 1, and a non-constant change in Y values - sometimes increasing by 1 and sometimes by 4
Therefore, according to the rule, the table does not describe a function
Answer:
No
Video Solution
Exercise #5
Determine whether the following table represents a constant function
Step-by-Step Solution
To determine if the table represents a constant function, we need to examine the Y-values corresponding to the X-values given in the table.
- Step 1: Identify the given values from the table. The pairs are as follows: - For , - For , - For ,
- Step 2: Check if all Y-values are the same. Compare Y-values for each X-value: - when , - when , - when .
Since the Y-values (2, 4, and 7) are not the same, the function is not constant.
Thus, the table does not represent a constant function. The correct choice is: No.
Answer:
No
Video Solution
Frequently Asked Questions
What are the 4 main ways to represent a function?
+Functions can be represented in four primary ways: algebraic (equations like f(x) = 2x + 3), graphical (coordinate plane plots), tabular (input-output value tables), and verbal (written descriptions of relationships).
How do I convert from algebraic to graphical representation of functions?
+To graph a function from its equation: 1) Choose at least 3 x-values, 2) Substitute each x-value into the equation to find corresponding y-values, 3) Plot these coordinate points on a coordinate plane, 4) Connect the points with a straight line.
What's the difference between Y= and f(x) notation?
+Both Y= and f(x)= represent the same concept - the output depends on the input x. Function notation f(x) is more formal and useful for evaluating specific values, while Y= is simpler for basic equations.
How can I tell if a function is increasing or decreasing?
+Two methods work: 1) Look at the coefficient of x in the algebraic form - positive means increasing, negative means decreasing, 2) Examine the graph from left to right - if it goes up, it's increasing; if it goes down, it's decreasing.
Why is each x-value paired with only one y-value in functions?
+This is the fundamental definition of a function - each input (x) must produce exactly one output (y). If one x-value gave multiple y-values, it wouldn't be a function but rather a relation.
How do I create a table of values from a function equation?
+Choose several x-values (usually including 0, positive, and negative numbers), substitute each into the equation, calculate the corresponding y-values, then organize the pairs in a two-column table with x and y headers.
What makes a good verbal representation of a function?
+A clear verbal representation describes the relationship between variables using everyday language. For example, 'the output is twice the input plus three' for f(x) = 2x + 3, or real-world contexts like 'each flour package makes 3 pizzas.'
Which function representation is best for solving problems?
+Each representation has strengths: algebraic for calculations, graphical for visualizing trends and intercepts, tabular for specific value lookups, and verbal for understanding real-world contexts. The best choice depends on the problem type.