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3.3 — Features of Functions
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1. In class: A Practice Understanding Task
Use these reminders as you describe the graphs, tables, and situations.
Does it represent a function?
Each input must have exactly one output. On a graph, use the vertical line test: no vertical line should meet more than one included point. Repeated outputs are fine; two different outputs for the same input are not. Open circles are not included.
Read the features
Domain: the x-values represented. Range: the y-values reached. Filled endpoints are included; open endpoints are excluded. For separated dots, list individual values in braces rather than filling the spaces with an interval.
Worked example 1 — Domain and range of separate points
Problem: The graph below consists of just three points. Find its domain and range.
Solution: The x-values are −2, 0, and 3, so the domain is \(\{-2,0,3\}\). The y-values are 1 and 4, so the range is \(\{1,4\}\). List 4 only once, even though two points have that output.
An interval includes every real number between its endpoints. So \([-2,3]\) would include inputs such as 1, 2, and 1.3, none of which belong to this graph. Even if every integer from −2 to 3 were included, we would still need to list those integers in braces; the interval would incorrectly include the fractional values between them. Likewise, \([1,4]\) would include outputs such as 2.5, but this graph’s only outputs are 1 and 4.
An x-intercept is a point where a graph meets the x-axis, so its y-coordinate is zero. A y-intercept is a point where a graph meets the y-axis, so its x-coordinate is zero. Write intercepts as ordered pairs.
A function is increasing where it rises from left to right, decreasing where it falls, and constant where it stays level. Describe these using intervals of x-values.
A relative maximum or minimum is a local peak or valley. An overall maximum or minimum is the greatest or least output on the entire graph. Give both the output and the input where it occurs.
Symmetry and end behavior
End behavior: describe what happens at the left and right ends—whether the graph rises, falls, or approaches a height.
2. Ready — Find intersections using tables
Fill value tables and find matching outputs at the same input.
Try Ready 1–4 yourself first, then use the answer key below to check your intersection points.
The intersection of two graphs has the same input and the same output in both functions. A pair of value tables locates it when both functions give matching outputs for one listed x-value. Comparing outputs from different rows does not locate an intersection.
Worked example 2 — Match outputs at the same input
Use tables to find the intersection of \(f(x)=2x-1\) and \(g(x)=-x+5\).
| x | f(x) = 2x − 1 | g(x) = −x + 5 |
|---|---|---|
| −1 | −3 | 6 |
| 0 | −1 | 5 |
| 1 | 1 | 4 |
| 2 | 3 | 3 |
For example, at \(x=-1\), \(f(-1)=2(-1)-1=-3\) and \(g(-1)=-(-1)+5=6\). At \(x=2\), both functions give 3, so the intersection is \(\boxed{(2,3)}\).
Check directly: \(2(2)-1=3\) and \(-2+5=3\). The shared input 2 and shared output 3 form the ordered pair.
Fill each packet table using the stated x-values, then look for the same output at the same input. Handle negative inputs carefully: subtracting a negative changes the sign, as in \(-(-1)=1\).
Check your work — Ready 1–4 answer key
- \((3,4)\)
- \((2,4)\)
- \((2,2)\)
- \((5,11)\)
3. Set — Evaluate claims about linear and exponential functions
Use examples and counterexamples to assess linear, exponential, and sequence claims.
A claim about all functions of a type must hold in every case. One counterexample is enough to show that it is false.
Check your work — Set 5–10 answer key
5. All linear functions are increasing.
False. A line increases when its slope is positive, decreases when its slope is negative, and is constant when its slope is zero. For example, \(y=-x+4\) decreases.
6. Arithmetic sequences are an example of linear functions.
True. An arithmetic sequence adds the same amount each time its index increases by 1. Its points lie on a straight line. For example, \(a(n)=5+2(n-1)\) gives \(5,7,9,\ldots\) for \(n=1,2,3,\ldots\). The domain consists of the sequence’s term numbers, so its graph is separate points on that line.
7. Exponential functions have a domain that includes all real numbers.
True for the unrestricted exponential functions used here, such as \(y=2^x\). Positive, zero, negative, and fractional inputs are all allowed. For example, \(2^0=1\) and \(2^{-1}=\frac12\).
8. Geometric sequences have a domain that includes all integers.
False. The inputs are the sequence’s term numbers, usually \(1,2,3,\ldots\), or sometimes \(0,1,2,\ldots\). Negative integers are not included in these sequences. A sequence and a continuous exponential function can have similar formulas but different domains.
9. The range for an exponential function includes all real numbers.
False. For example, \(y=2^x\) has only positive outputs, so its range is \((0,\infty)\). It never gives zero or a negative number. This single example disproves the claim.
10. All linear relationships are functions with a domain and range containing all real numbers.
False. A vertical line such as \(x=2\) is a linear relationship but is not a function of x: the same x-value has many y-values. Also, a horizontal line such as \(y=5\) is a function, but its range is only \(\{5\}\). A nonconstant function \(y=mx+b\) with unrestricted inputs does have all real numbers in its domain and range.
4. Go — Read the domain from a graph
Read each graph’s domain and write it in interval notation.
For a refresher, see Section 3.2’s explanation of interval notation and endpoints. Read the domain from the graph’s x-values, including or excluding endpoints as shown. Try Go 11–14 independently, then use the answer key below to check your work.
Check your work — Go 11–14 answer key
- \([-4,0]\cup[1,3]\cup[4,5]\)
- \((-\infty,\infty)\)
- \((-\infty,3)\)
- \((0,\infty)\)
Additional instructional resources
| Resource | Use it for | Scope |
|---|---|---|
| Khan Academy — Worked example: domain and range from graph | Reading horizontal and vertical bounds. | Close match for Main task graph features and Go. |
| The Organic Chemistry Tutor — Domain and Range of a Function From a Graph | Read the included inputs and outputs and express them in interval notation. | Supports the in-class graph features and Go 11–14. Use braces for the separate-point domains discussed in this lesson. |
| Khan Academy — Function symmetry introduction | Illustrated introduction to y-axis and origin symmetry. | Use the graph illustrations to learn to recognize symmetry. The article also introduces the terms “even” and “odd”; those terms are not required for this packet. |
An internet connection is required for MathJax equation rendering and external resources. The original diagrams are embedded in this file.
Teacher’s review of the source packet
No serious errors require a special warning here. The key instructional distinctions are one output per input, relative versus overall extrema, and continuous inputs versus separate sequence indices. Tables and sketches are read using their intended classroom patterns.
Coverage — Section 3.3
| Packet component | Skills taught | Where it is taught |
|---|---|---|
| Main 1–16 · pp. 11–13 | Reminders for the function test, graph features, symmetry, and end behavior | In-class reference, separated-points domain-and-range example, and a clearly flagged symmetry prerequisite with an external learning reference. |
| Ready 1–4 · p. 14 | Evaluate two tables and identify equal outputs at a common input | Section 2; original table and substitution checks. |
| Set 5–6 and 10 · p. 15 | Linear slope cases, arithmetic sequences, constant and vertical lines | Section 3, answer key for problems 5, 6, and 10. |
| Set 7–9 · p. 15 | Exponential curve domains/ranges versus geometric sequence indices | Section 3, answer key for problems 7–9. |
| Go 11–14 · p. 15 | Domain intervals, open/closed endpoints, arrows, gaps and unions | Section 4; domain reminder, link to Section 3.2’s interval explanation, and answer key. |
Source: Mathematics Vision Project, Secondary Mathematics One, Module 3: Features of Functions (May 2016), Section 3.3, printed pp. 11–15, and its teacher notes. Worked examples and diagrams are original. This lesson uses the preceding sections’ graph and sequence skills without introducing the function operations or algebraic system-solving methods of later sections.
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