decide whether a relationship is a function, use function notation to evaluate and solve, and interpret statements in function notation in context.
Instruction
Much of algebra is about how one quantity depends on another. A function is a rule that assigns to each input exactly one output. The set of allowed inputs is the domain; the set of outputs the function actually produces is the range.
Function or not? “Each student’s height on the first day of school” is a function of the student: each student has one height. “The students who are 160 cm tall” is not a function of height: the input 160 could give several students. A table represents a function if no input appears with two different outputs. A graph represents a function if every vertical line crosses it at most once, because a vertical line holds one input fixed; two crossings would mean two outputs for that input.
Function notation. Streaming video drains a phone battery. Suppose (illustratively) a full battery drops 12.5 percentage points per hour. Name the function B: B(t) is the battery percentage after t hours, and
B(t) = 100 − 12.5t
Read B(t) as “B of t“, not “B times t“. The parentheses hold the input.
- Evaluate: B(3) = 100 − 12.5(3) = 62.5. In context: after 3 hours of streaming, the battery is at 62.5%.
- Solve: when is the battery at half? Set B(t) = 50: 100 − 12.5t = 50, so 12.5t = 50 and t = 4. After 4 hours.
- Interpret a statement: B(8) = 0 says that after 8 hours the battery is empty.
Notice the difference between evaluating and solving. Evaluating gives you an input and asks for the output. Solving gives you an output and asks which input produces it.
Domain and range in context. The formula 100 − 12.5t can take any number as input, but the situation cannot. Time cannot be negative, and after 8 hours the battery is dead, so a sensible domain is 0 ≤ t ≤ 8. The range is then 0 ≤ B ≤ 100. The graph is a line segment from (0, 100) to (8, 0), not a full line. Every point on it is a solution of the equation y = 100 − 12.5t, which is what it means for this to be the graph of B.
Why the “exactly one” rule matters. If an input could give two outputs, B(3) would be ambiguous: which battery level do you mean? The one-output rule is what makes notation like B(3) meaningful and lets you use a model to make a single prediction.
Try it. Classify each relationship on the card set as a function or not, with a reason. Then, for the function C(n) = 4 + 1.5n, the cost in dollars of a burrito with n extra toppings, evaluate C(2), solve C(n) = 10, and state a sensible domain.
Formative check
Work through these before moving on. They are not graded — they tell you, and your teacher, whether the standard below has landed yet.
For B(t) = 100 − 12.5t, what does B(6) = 25 mean?
A table in which the input 3 appears twice, once with output 5 and once with output 7, represents a function.
For C(n) = 4 + 1.5n, C(2) = , and C(n) = 10 when n = .
Give one relationship from your own life that is a function and one that is not, explaining each with the one-output rule. For the function, state a sensible domain and range with units.
How confident are you that you can decide whether a relationship is a function and use function notation to evaluate, solve and interpret?
Practice
Work these on paper or in your notebook, then open Check your answer. Aim for all of Fluency and Application; try at least one Challenge.
Printable version: this unit’s practice workbook (PDF)
Fluency
Build speed and accuracy with the core skill.
- If f(x) = −2x2 + 2, find f(1).
Check your answer
Answer: f(1) = 0Substitute, then follow the order of operations; square the input first. - If f(x) = x2 − 2x − 9, find f(−4).
Check your answer
Answer: f(−4) = 15Substitute, then follow the order of operations; square the input first. - If f(x) = −6x − 1, find f(−3).
Check your answer
Answer: f(−3) = 17Substitute, then follow the order of operations; square the input first. - If f(x) = x2 − x − 6, find f(−5).
Check your answer
Answer: f(−5) = 24Substitute, then follow the order of operations; square the input first. - If f(x) = 3x2 − x, find f(3).
Check your answer
Answer: f(3) = 24Substitute, then follow the order of operations; square the input first. - If f(x) = x2 − 2x + 2, find f(1).
Check your answer
Answer: f(1) = 1Substitute, then follow the order of operations; square the input first. - For B(t) = 100 − 12.5t, find B(5) and solve B(t) = 25.
Check your answer
Answer: B(5) = 37.5; t = 6Evaluate: substitute. Solve: set the output to 25. - For C(n) = 4 + 1.5n, find C(3) and solve C(n) = 11.5.
Check your answer
Answer: C(3) = 8.5; n = 5
Application
Use the skill in context. Show your reasoning.
- A transit fare card starts with $40 and each ride costs $2.50 (illustrative): f(r) = 40 − 2.5r. Find f(6), and state a sensible domain and range.
Check your answer
Answer: f(6) = 25; domain: whole numbers 0 ≤ r ≤ 16; range: 0 ≤ f ≤ 40 in steps of $2.50The card is empty after 40 ÷ 2.5 = 16 rides. - Does the table {(1, 3), (2, 5), (1, 7), (4, 9)} represent y as a function of x? Explain.
Check your answer
Answer: No: the input 1 has two outputs, 3 and 7. - T(h) is the temperature in °F in Fresno h hours after midnight on a July day. Interpret T(14) = 96, and say what T(h) = 80 asks.
Check your answer
Answer: At 2 p.m. the temperature was 96°F. T(h) = 80 asks at what time(s) the temperature was 80°F.
Challenge
Stretch problems. Expect to think before you write.
- For f(x) = 3x − 7, find a so that f(a) = f(2a) − 12, and check.
Check your answer
Answer: a = 4: f(4) = 5 and f(8) − 12 = 53a − 7 = 6a − 19 gives a = 4.
Review
Keep earlier skills sharp.
- Solve: 2x + 7 = −x − 14
Check your answer
Answer: x = −7Move the x-terms to one side: 3x = −21. - For the data 4, 5, 7, 8, 8, 16, 16, 23, 28, 40: find the mean, median, range and interquartile range. (Find the quartiles as the medians of the lower and upper halves, leaving out the overall median when the count is odd.)
Check your answer
Answer: mean = 15.50; median = 12.0; range = 36; IQR = 16.0 (Q1 = 7.0, Q3 = 23.0)