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 graph is a function, we will use the Vertical Line Test.
The Vertical Line Test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.
Let's apply this test to the given graph, where a horizontal line is drawn. This line represents the function the graph should be verified against.
- Step 1: Conceptualize vertical lines passing through different x-values across the domain of the graph.
- Step 2: Observe if any of these vertical lines intersect the graph at more than one point.
Upon inspection of the graph, we see that every vertical line intersects the graph at exactly one point.
This indicates that for every input (x-value), there is a unique output (y-value), fulfilling the criteria for the definition of a function.
Therefore, according to the Vertical Line Test, the given graph is indeed a function.
The correct choice is: Yes
Answer:
Yes
Video Solution
Exercise #2
Is the given graph a function?
Step-by-Step Solution
To determine whether the graph represents a function, we apply the Vertical Line Test. Here are the steps we follow:
- Step 1: Visualize placing a vertical line across various parts of the graph.
- Step 2: Check if the vertical line intersects the graph at more than one point at any given position.
Step 1: On evaluating the given graph carefully, there is a notable presence of a vertical line passing through multiple y-values. Specifically, the vertical line goes from to at .
Step 2: Since this vertical line at intersects the graph at an infinite number of points, it fails the Vertical Line Test.
Therefore, the graph does not represent a function. According to our analysis and the Vertical Line Test, the correct answer is No.
Answer:
No
Video Solution
Exercise #3
Is the given graph a function?
Step-by-Step Solution
To determine if the graph in question represents a function, we'll employ the Vertical Line Test. This test helps to ascertain whether each input value from the domain (x-values) is connected to a unique output value (y-values).
- According to the Vertical Line Test, a graph represents a function if no vertical line can intersect the graph at more than one point.
- In the provided diagram, the graph is a straight line.
- Visual inspection shows that any vertical line drawn at any point along the x-axis intersects the line exactly once.
- This indicates that for each x-value, there is a unique corresponding y-value. Therefore, the relationship depicted by the graph meets the criteria for a function.
Thus, the given graph correctly characterizes a function.
Therefore, the solution to the problem is Yes.
Answer:
Yes
Video Solution
Exercise #4
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 #5
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
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.