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Algebra I – Functions and Variations

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    Study Guide Algebra I – Func Ɵ ons and Varia Ɵ ons 1. Func Ɵ ons 1. Understanding Relations First Before we learn about functions , we need to understand something called a relation . A relation is simply a set of ordered pairs. An ordered pair looks like this: [(x, y)] The first number is the x - value , and the second number is the y - value . So, if we have: [A = {( - 1,1), (1,3), (2,2), (3,4)}] This set of ordered pairs is called a relation . “Figure 1 shows a graph of the ordered pairs ( - 1,1), (1,3), (2,2), (3,4).”

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    Study Guide 2. Domain and Range Now let’s talk about two very important ideas: Domain The domain is the set of all x - values. Range The range is the set of all y - values. From the example: [A = {( - 1,1), (1,3), (2,2), (3,4)}] • Domain = { − 1, 1, 2, 3} • Range = {1, 2, 3, 4} Notice: • We list each number only once. • Order does not matter in sets. Example Find the domain and range of the plotted points. Domain = { − 2, − 1, 1, 3} Range = { − 1, 2, 3}

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    Study Guide “ Figure 2 Plotted points.” 3. What Makes a Relation a Function? Now we move to something more specific: functions . A function is a special type of relation. Definition of a Function A function is a relation where: Each x - value is paired with exactly one y - value. That means: • No x - value can have two different y - values. • Each input gives only one output. 4. Example of a Function Consider the equation: [y = x + 1]

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    Study Guide This is a function because: • For every value of x, • There is exactly one value of y. For example: • If x = 2, then y = 3 • If x = − 1, then y = 0 There is never more than one answer for y. The domain and range of this function are: • All real numbers “ Figure 3. A graph of the linear equation y = x + 1.” 5. Graphs of Functions Let’s look at some more examples of functions. (a) ( y = |x|)

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    Study Guide This graph forms a V - shape. Each x - value gives exactly one y - value. (b) ( y = x 2 ) This graph is a parabola. Again, every x has only one y. (c) ( y = sin x) This graph is a wave pattern. Still, each x corresponds to one y - value. All of these are functions. “ Figure 4 Graphs of functions.” 6. Graphs That Are NOT Functions Now let’s see what is not a function. A relation is not a function if: A single x - value is paired with two or more y - values.

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    Study Guide Let’s examine these cases: (a) x = 2 This is a vertical line. The same x - value (2) has many y - values. Not a function. (b) (x 2 + y 2 = 4) This is a circle. Some x - values match with two y - values. Not a function. (c) (x = y 2 ) This sideways parabola gives two y - values for some x - values. Not a function. • x = 2 • circle (x 2 + y 2 = 4) • (x = y 2 )

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    Study Guide “ Figure 5 Graphs of relations that are not functions.” Example 2: Understanding Domain, Range, and Functions What Is a Relation? A relation is simply a set of ordered pairs. For example: [B = {( - 2,3), ( - 1,4), (0,5), (1, - 3)}] Each ordered pair has: • A first number → this belongs to the domain • A second number → this belongs to the range What Is the Domain? The domain is the set of all x - values (first numbers). From set B: Domain = { − 2, − 1, 0, 1} Think of the domain as:

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    Study Guide “What values go INTO the function?” What Is the Range? The range is the set of all y - values (second numbers). From set B: Range = { − 3, 3, 4, 5} Think of the range as: “What values come OUT of the function?” When Is a Relation a Function? A relation is a function if: Each x - value is paired with only one y - value. If one x - value has two different y - values, it is not a function. Looking at Graphs Now let’s understand what the graphs are showing. 1 & 2 — Checking Domain and Range You are given sets and asked to: • Identify the domain • Identify the range • Decide if it is a function

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    Study Guide If each x appears only once → Function If an x repeats with different y - values → Not a function 3 — Graph with Several Points Look at the plotted points carefully. To find the: • Domain → look at all x - coordinates • Range → look at all y - coordinates To check if it’s a function: Use the Vertical Line Test . If a vertical line touches the graph at more than one point → Not a function.

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    Study Guide 4 — Range Given as an Inequality Range: { y : y ≥ − 3 } This means: All y - values are greater than or equal to − 3. If every x gives exactly one y → it is a function. 5 — A Line with an Arrow The graph shows: • A solid point at (0,0) • A line extending upward • A point at (1, − 3) Domain is written as:

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