Domain & Range
Notation Station
Function or Not?
Build the Line
Models in Motion
100
A ray starts at an OPEN circle at (-4, -2) and rises forever up and to the right. Write the DOMAIN in set notation.
{x ∈ ℝ | x > -4}
The circle is open, so -4 is excluded. The arrow means no right-hand bound.
100
Rewrite {x ∈ ℝ | x > -4} in INTERVAL notation.
(-4, ∞)
Strict inequality gives a parenthesis, and infinity always gets a parenthesis.
100
Is {(1, 2), (1, 5), (3, 4)} a function? Say why.
No - the input 1 produces two different outputs, 2 and 5. A function allows exactly one output per input.
100
Mayerlin starts with $35 and adds $10 every week. Write her savings function.
M(x) = 10x + 35
The "per" number is the slope; the starting amount is the y-intercept.
100
The value of a car t years after purchase is V(t) = 28000 - 4000t dollars. Find V(4) and write a sentence saying what it means.
V(4) = 12000. "The value of the car after 4 years is $12,000." The number alone is half credit. The sentence with units is the other half.
200
A parabola opens upward with vertex (-2, 1) and arrows on both ends. Write the DOMAIN and RANGE in interval notation.
Domain (-∞, ∞) Range [1, ∞)
Arrows on both ends accept every input. The vertex is the lowest point and it is included, so the range gets a bracket at 1.
200
Rewrite [-5, ∞) in SET notation.
{x ∈ ℝ | x ≥ -5}
The square bracket means -5 is included, so it becomes ≥.
200
Name the test for deciding whether a GRAPH is a function, and state what a failure looks like.
The vertical line test - if any vertical line crosses the graph more than once, it is not a function. The test is just the definition, drawn.
200
A table shows x = 4, 8, 12, 16 with y = $35, $65, $95, $125. Find the SLOPE.
m = 30/4 = 15/2 = 7.5 dollars per week
Δy ÷ Δx using any two rows, then check it is the same for every pair.
200
The value of a car t years after purchase is V(t) = 28000 - 4000t dollars. Find t when V(t) = 8000, and say what it represents.
t = 5. "The car is worth $8,000 after 5 years."
Given an output, you solve. Given an input, you substitute.
300
A segment runs from a CLOSED circle at (-3, -5) to an OPEN circle at (4, 6). Write the DOMAIN and RANGE in set notation.
{x ∈ ℝ | -3 ≤ x < 4} {y ∈ ℝ | -5 ≤ y < 6}
Closed on the left is included, open on the right is excluded. Mixed endpoints are the whole point of this one.
300
A student wrote [-∞, 5]. What is wrong with it, and what is the correct answer?
Infinity can never be reached, so it never gets a square bracket.
Correct: (-∞, 5]
300
Is x² + y² = 9 a function? Justify with a specific numerical example.
No - for example x = 0 gives both y = 3 and y = -3. The example is what earns the mark. "It's a circle" on its own does not.
300
Same table (x = 4 → $35, x = 8 → $65, ...). Write the equation in SLOPE-INTERCEPT form. Careful.
y = (15/2)x + 5
The trap: the table starts at week 4, so $35 is NOT the intercept. Point-slope with (4, 35) gives b = 5.
300
The value of a car t years after purchase is V(t) = 28000 - 4000t dollars. Give a REASONABLE DOMAIN, with units, and show how you found the upper end.
[0, 7] years - found by solving 28000 - 4000t = 0, so t = 7. Reasonable means it makes sense in the real world. Time starts at 0; the model dies when the car is worth $0.
400
A circle of radius 3 is centered at the origin. Write the DOMAIN and RANGE in interval notation.
Domain [-3, 3] Range [-3, 3]. The circle reaches -3 and 3 in both directions, and both edges are included.
400
Write "all real numbers except 2" in interval notation AND in set notation.
(-∞, 2) ∪ (2, ∞) and {x ∈ ℝ | x ≠ 2}
A single missing value splits the set into two pieces joined by a union.
400
For which value(s) of a is the set {(−2, 5), (0, 1), (a, 7)} not a function? Explain.
a = −2 or a = 0
400
A line passes through (0, 40) and (10, 165). Write the function.
A(x) = 12.5x + 40
Slope = (165 - 40) / (10 - 0) = 12.5, and (0, 40) hands you the intercept directly.
400
A balloon is at 50 ft and rises at 12 ft/s for 5 seconds. How high is it at t = 5, and what is the slope-intercept equation for the climb?
110 ft. y = 12x + 50 50 + 12(5) = 110. That turning point is what the descent equation is built from.
500
A curve has a VERTICAL asymptote at x = 1 and a HORIZONTAL asymptote at y = 0, with a branch on each side. Write the DOMAIN and RANGE in interval notation.
Domain (-∞, 1) ∪ (1, ∞) Range (-∞, 0) ∪ (0, ∞).
A vertical asymptote punches a hole in the domain; the horizontal one punches a hole in the range. Both need a union and parentheses.
500
Convert {y ∈ ℝ | -2 < y ≤ 7} to interval notation, then state which endpoint is included and which is not.
(-2, 7] - 7 is included, -2 is not. Each endpoint is decided separately. Mixed brackets are normal and correct.
500
Can a function have TWO DIFFERENT INPUTS with the SAME OUTPUT? Answer yes or no, and give an example.
Yes. For example y = x² gives f(2) = 4 and f(-2) = 4, and it is still a function. The rule restricts outputs per input, not inputs per output. Repeated y-values are fine; repeated x-values are not.
500
A line has slope -20 and passes through (5, 110), which is NOT the y-intercept. Write it in POINT-SLOPE form, then say why that form is the right choice.
y - 110 = -20(x - 5)
We know the slope and a point that is not the y-intercept, which is exactly when point-slope is used. Slope + intercept → y = mx + c; slope + any other point → y - y₁ = m(x - x₁).
500
A balloon is 50 ft above the ground at t = 0. It rises at 12 ft/s for 5 seconds, reaching (5, 110), then descends at 20 ft/s. How long until it hits the ground?
10.5 seconds after it was at 50 ft, which is 5.5 seconds after the descent began.
0 - 110 = -20(x - 5), so x = 10.5.