Math 112 Precalculus Mathematics - Functions and Graphs (Flashcards)
Functions and Graphs: Comprehensive Notes (MATH 112 - Precalculus Mathematics)
This note compiles key ideas, definitions, tests, and example problems drawn from the provided transcript. It emphasizes definitions, procedures, and common problem types you’ll encounter in the course.
1) What is a Function? Basic Concepts
A function is a fundamental mathematical concept that describes a specific relationship between two sets of values. It is a relation that assigns to every input (from a set called the domain) exactly one output (to a set called the range). This 'exactly one' rule is crucial distinguishing a function from a general relation.
Notation: If x represents an input value, the unique output value produced by the function for that input is denoted as f(x). This is read as "f of x" and is often used interchangeably with y, so we write y = f(x). The letter f is a common choice for function names, but other letters like g or h can also be used.
In a table, which often lists pairs of (x, f(x)) or (x, y) values:
The domain is the set of all unique input x-values listed in the table. These are the values for which the function is defined.
The range is the set of all unique output f(x)-values (or y-values) produced by the function. These are the results obtained from applying the function to its inputs.
Important distinction: For a relation to be classified as a function, it must satisfy the condition that no single input x-value maps to more than one output y-value. If you find an x-value that is paired with two or more different y-values, the relation is not a function.
Example (from transcript, Problem 1 table):
Given x values: (x = -4,-3,-2,-1,0,1,2,3) with corresponding f(x): (4,1,0,-4,1,-1,3,-3).
(a) To find f(x) = 1, we look for which x-values produce an output of 1. From the table, this occurs at x = -3 and x = 0. (Note: Multiple inputs can map to the same output, and it still constitutes a function).
(b) To find f(x) = 5, we scan the f(x) values for 5. There is no output equal to 5 in the table, so there is no solution (i.e., no x gives y = 5).
(c) To find f(0), we locate the input x = 0 in the table. The corresponding output is 1. Thus, f(0) = 1.
(d) To find when f(x) \leq 0, we identify all x-values whose corresponding f(x) is less than or equal to zero. These are x = -2 (where f(x)=0), x = -1 (where f(x)=-4), x = 1 (where f(x)=-1), and x = 3 (where f(x)=-3).
2) Determining If a Relation Is a Function
From a table: To check if a relation represented in a table is a function, visually inspect the column (or row) of input x-values. If any x-value appears more than once, check its corresponding y-values (or f(x)-values). If an x-value repeats with different y-values, then the relation is not a function. If an x-value repeats but always with the same y-value, it is still a function.
From a graph: The Vertical Line Test (VLT) is a simple graphical method to determine if a graph represents a function. Imagine drawing vertical lines across the entire graph. If any vertical line intersects the graph at more than one point, then the graph does not represent a function. This is because multiple intersection points on a single vertical line would mean a single x-value has multiple y-values, violating the definition of a function.
Examples from transcript (Problem 2):
Table 1: Let's assume this table has no repeated x-values, or if x-values repeat, the corresponding y-values are always the same. Therefore, it describes a function.
Table 2: This table explicitly states it's not a function because the input x = 1 maps to both y = -3 and y = 9. This violates the unique output requirement for a function.
Table 3: Similar to Table 1, if all x-values either appear once or map to the same y-value if they repeat, then it represents a function.
Graph-based examples (Problems 3–7):
Graph 1 passes the vertical line test: Any vertical line drawn across this graph intersects it at most once. Therefore, y is a function of x.
Graph 2 fails the vertical line test: You can draw at least one vertical line that intersects the graph at two or more points. Thus, it is not a function of x.
General rule: If two or more points lie on the same vertical line in a relation, it cannot be a function, as it implies a single x-input yields multiple y-outputs. The only exception would be if those 'points' are actually coincident (the same point), but distinct points along a vertical line immediately disqualify it as a function.
3) Functions From Equations and Graphs
To determine if an equation defines y as a function of x, the general strategy is to solve the equation for y in terms of x. After solving, if for every valid x-value, you get exactly one corresponding y-value, then y is a function of x. If you find that a single x-value can produce two or more y-values (e.g., due to a \pm sign from a square root, or an even power of y), then y is not a function of x.
1) 2|x| + y = 4: Solve for y: y = 4 - 2|x|. For any given x, the absolute value |x| is unique, and thus 4 - 2|x| produces a single, unique y-value. Therefore, this is a function.
2) x^2 + 2y = 7: Solve for y: 2y = 7 - x^2 \rightarrow y = \frac{7 - x^2}{2}. For any x, x^2 is unique, and so is \frac{7-x^2}{2}. Thus, this is a function.
3) 2x = y^2: Solve for y: y = \pm\sqrt{2x}. Because of the \pm sign, for any positive x (that makes 2x positive), there will be two possible y-values. For example, if x=2, y = \pm\sqrt{4} = \pm2. Therefore, this is not a function.
4) x^2 + (y-2)^2 = 16: This is the equation of a circle centered at (0,2) with a radius of 4. To solve for y: (y-2)^2 = 16 - x^2 \rightarrow y-2 = \pm\sqrt{16 - x^2} \rightarrow y = 2 \pm\sqrt{16 - x^2}. Again, the \pm sign indicates two possible y-values for most x-values in the domain of the square root (which is [-4,4]). Therefore, this is not a function in general.
Summary rule: When converting an equation to solve for y in terms of x, if you end up with a single, unambiguous expression for y for each x, it’s a function. If you must use a \pm sign or obtain multiple distinct values for y for a single x, then it’s not a function. Graphically, these non-functions often fail the Vertical Line Test.
4) Graphical Tests and Function Properties
The Vertical Line Test (VLT) is the definitive graphical test for functions: a graph represents a function if and only if every vertical line drawn through its domain intersects the graph at most once. If even one vertical line crosses the graph at two or more points, it signifies that a single x-input corresponds to multiple y-outputs, thereby disproving its function status.
Problem-type takeaway: If a graph clearly shows that a vertical line (e.g., x = c) intersects the graph at more than one point, then the graph does not represent a function. This is a common visual cue to identify non-functions.
For piecewise graphs, which are composed of different function pieces defined over different parts of the domain, you must perform the VLT on the entire graph. Special attention should be paid to the endpoints where the pieces meet. If an endpoint of one piece is an open circle and the corresponding endpoint of another piece (at the same x-value) is a closed circle, and they align vertically, it can still be a function. However, if two closed circles exist directly above/below each other at a boundary x-value, or if overlapping pieces create multiple y-values for a single x, it will fail the VLT.
Example (Problem 4): Consider a graph that fails the VLT because, for instance, at x = 2, there are points P and Q, and at x = 5, there are points R and S. To make this relation a function, you would need to remove one point from each vertical line that fails the VLT. The specific choice given, “Q and R,” means removing Q from x=2's vertical line and R from x=5's vertical line, ensuring only one point remains on each of those problematic vertical lines. This process modifies the relation to pass the VLT.
Example (Problem 5): Graph 1 passes VLT and thus represents a function. Graph 2 does not pass VLT (at multiple x-values, a vertical line would cross it twice) and therefore does not represent a function.
5) Function Values from Graphs and Tables
When evaluating function values from a graph, such as finding f(-1) or g(4), you locate the given input x-value on the horizontal (x) axis. Then, move vertically from that x-value until you intersect the graph of the specified function (e.g., f or g). The corresponding y-value at that intersection point on the vertical (y) axis is the output of the function.
For instance, if a graph shows f (often red) and g (often blue) on the same axes:
To find f(-1), locate x = -1 on the horizontal axis, then move up or down to intersect the red graph. Read the y-value at that point.
To find g(-1), perform the same process but intersect the blue graph.
To compute f(-1)+g(-1), you would add the separately read y-values together.
If only a blue section is visibly graphed over a red section for example, the problem statement often clarifies whether the red function is considered