Function notation f(x), domain & range

Worked example

4.2.w1 Example 1: f(x)=3x-1 at several inputs. $$f(2)=3(2)-1=6-1=5,\quad f(0)=3(0)-1=-1,\quad f(-2)=3(-2)-1=-6-1=-7.$$ Look at the parentheses around the −2. Writing 3(−2) keeps the sign attached, so it becomes −6 and the answer lands on −7. That's the payoff for the fussy parentheses on the easy cases.

4.2.w2 Example 2: "not multiplication." Suppose f(2) gets read as f·2, as if f were a number to multiply by. There is no number f here; f is just the name of the recipe, and f(2) means the recipe's output at 2. Run it properly: 3(2) − 1 = 5. Reading it aloud as "f of 2" is the quickest cure.

4.2.w3 Example 3: g(x)=x²+1. $$g(3)=3^2+1=9+1=10,\quad g(0)=0+1=1,\quad g(-2)=(-2)^2+1=4+1=5.$$ The one to slow down on is g(−2). The parentheses around −2 mean you square the whole input: (−2)² is (−2)(−2) = +4, not −4. So g(−2) comes out to 5.

4.2.w4 Example 4: evaluate at an expression. With f(x)=3x-1: $$f(2a)=3(2a)-1=6a-1.$$ Whatever sits in the parentheses goes everywhere x was. Here 2a takes x's place, so 3(2a) is 6a.

4.2.w5 Example 5: domain & range from a table. For

the Domain is the input row, \(\{1,2,3,4\}\). The Range is the outputs with repeats listed once: \(\{5,7,9\}\). The output 5 happens at both x=1 and x=2, but you write it a single time, because the range is just the set of values that come out.

4.2.w6 Example 6: discrete vs. continuous domain (why braces for one, "all reals" for the other). This is the difference between Example 5's table and the line f(x)=3x−1.

• A table or pair-list gives a discrete domain: a handful of separate inputs. You plot them as separate dots and list them in braces. Concrete case: tickets cost $12 each, so the cost of n tickets is t(n)=12n. You can buy 0, 1, 2, 3, … tickets but never 2.5, so the domain is the separate whole numbers {0,1,2,3,…}: separate dots, don't connect them.
• A line like f(x)=3x−1 accepts every real number in between, with no gaps, so its domain is continuous: an unbroken line, described in words as "all real numbers." Concrete case: a candle's height after t hours, for any t from 0 to 5; time can be 2.5 hours or 2.501 hours, so the inputs fill the whole range and the graph is an unbroken curve.

So the form of the answer follows the domain: a discrete domain → list in braces {…}; a continuous domain → say "all real numbers" (or "all real numbers from 0 to 5"). The one-line check: can the input land between two of your values? If no (tickets), it's discrete dots; if yes (hours), it's a continuous line. (No interval notation yet. That's later; words are enough here.)

After all that, here's a clean one to get the rhythm back before the practice mixes things up. With f(x) = 3x − 1, find f(1): substitute to get 3(1) − 1 = 3 − 1 = 2. One input, one substitution, done. That's the move underneath every evaluation in this lesson.

With a few clean evaluations behind you, look at the trickiest spot. When a negative goes into a squared term, the parentheses are everything. Reading g(−2) without them invites −2², which a lot of people would compute as −4, squaring the 2 and leaving the minus outside. But the input is the whole −2, so it's (−2)², and a negative times a negative is positive: +4. If you write the parentheses every time you substitute, this slip mostly can't happen. Substitute, then compute, in that order.