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3.5 — Pooling it Together
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1. In class: A Solidify Understanding Task
What does it mean to add two functions?
Interpret the combined amount and add the two graph heights at the same input.
In Section 3.4, Aly and Dayne each drained a pool at the water park. Let \(x\) be the number of minutes since they began. The amounts of water remaining, in gallons, are
Aly starts with 28,000 gallons and removes 1,400 gallons each minute; Dayne starts with 24,000 and removes 1,000 each minute. If we want the total amount remaining in the two pools, we add their amounts at the same time:
This is called the sum of the functions. The input is still time, but the output is now the total water amount. For each input, find the two outputs and add them.
Worked example 1 — Find the total after 5 minutes
First find the amount in each pool at \(x=5\):
Then add those amounts:
After 5 minutes, the two pools contain 40,000 gallons altogether. The sum graph contains \((5,40{,}000)\). Keep the input 5: we are adding two amounts measured at the same time.
The sum is also a function: each time has one amount in Aly’s pool and one in Dayne’s, so adding them gives exactly one total.
Build the sum graph from the two original graphs
Plot each combined output above the same time on the horizontal axis.
Choose a time on the horizontal axis. Read the height of Aly’s graph and the height of Dayne’s graph there. Add the two heights, then plot the total above that same time. Repeat at several useful times.
Worked example 2 — Make a table of combined amounts
| Time x (minutes) | Aly: a(x) (gallons) | Dayne: d(x) (gallons) | Total: g(x) (gallons) |
|---|---|---|---|
| 0 | 28,000 | 24,000 | 52,000 |
| 5 | 21,000 | 19,000 | 40,000 |
| 10 | 14,000 | 14,000 | 28,000 |
| 15 | 7,000 | 9,000 | 16,000 |
| 20 | 0 | 4,000 | 4,000 |
Each row uses one time. For example, at 10 minutes, each pool has 14,000 gallons, so the combined graph is at \((10,28{,}000)\).
Both pumps remove a constant amount per minute while both pools are draining, so the total also decreases at a constant rate. Its graph is a straight line. Two well-spaced points, such as \((0,52{,}000)\) and \((20,4000)\), determine that line; the other points let us check it.
The new graph’s height is a sum of the original heights. It is not halfway between the two graphs. While both original amounts are positive, their total is higher than either individual amount.
Find the sum by adding the equations
Add the two rules and connect the result to the table and graph.
We can find a rule for the total without calculating every row separately. Substitute the two draining rules into \(g(x)=a(x)+d(x)\), then combine like terms.
Worked example 3 — Write the equation for the total
The initial total is 52,000 gallons. While both pumps are operating, they remove \(1400+1000=2400\) gallons per minute altogether.
Check the rule at 5 minutes: \(g(5)=52{,}000-2400(5)=40{,}000\), the same total we found by adding \(a(5)\) and \(d(5)\).
The equation and graph agree because both methods do the same thing: add the two outputs at each input. You may also see the notation \((a+d)(x)\); it means the same thing as \(a(x)+d(x)\).
For two linear rules, the sum adds their starting values and their rates of change. That is why the sum of these two lines is another line. For other kinds of functions, still add the outputs at the same input, but the resulting graph need not be straight.
Describe the combined function’s features
Interpret the total’s domain, range, direction, and extreme values.
To add two restricted functions, use inputs where both functions are defined. Aly’s original draining rule applies from 0 through 20 minutes, and Dayne’s from 0 through 24. Their shared interval is \([0,20]\).
- Domain: \([0,20]\) minutes.
- Range: \([4000,52{,}000]\) gallons. Write the smaller amount first, even though the total is decreasing.
- Direction: decreasing, by 2,400 gallons per minute.
- Maximum: 52,000 gallons at \(x=0\).
- Minimum on this interval: 4,000 gallons at \(x=20\), when Aly’s pool has just become empty.
The unrestricted linear equation \(g(x)=52{,}000-2400x\) has all real numbers as its domain and range. The pool situation imposes limits on that equation. We must pay attention to those limits when interpreting its intercepts.
Why y-intercepts add but x-intercepts do not
Add vertical-intercept values and find the sum’s zero separately.
A y-intercept is a point where the graph meets the y-axis, so its input is 0. Both original y-intercepts therefore use the same input, and we can add their output values:
The new y-intercept is \((0,52{,}000)\). We add the water amounts and keep the time at zero.
An x-intercept is a point where the graph meets the x-axis. Aly’s graph reaches zero at 20 minutes, and Dayne’s at 24 minutes. Adding those times would give 44, but it does not tell us when their combined output is zero. To find an x-intercept of the sum equation, set its total output to zero.
Worked example 4 — Find the zero of the sum equation
The unrestricted sum line has x-intercept \(\left(\frac{65}{3},0\right)\). This is different from adding the original x-intercept inputs, \(20+24\).
More generally, a sum can be zero because a positive output and a negative output cancel. For example, if \(f(3)=2\) and \(h(3)=-2\), then \((f+h)(3)=0\), even though neither original output is zero.
For the two separate pools shown, Aly is empty after 20 minutes. Dayne still has 4,000 gallons, so from then on the total water remaining is simply Dayne’s amount. It reaches zero at 24 minutes. Continuing Aly’s original straight line past 20 would give negative water and an incorrect physical total.
If the intention is instead that both pumps keep drawing from one shared supply, then \(52{,}000-2400x\) models the total until \(x=\frac{65}{3}\). For that model, the domain is \(\left[0,\frac{65}{3}\right]\), the range is \([0,52{,}000]\), and the minimum is zero. State which situation you are using when giving the domain or emptying time.
The key distinction is that y-intercepts all use the input 0, whereas the original x-intercepts usually occur at different inputs. Function addition always combines outputs at the same input.
2. Ready — Find outputs, find inputs, and recover a rule
Evaluate functions, find inputs from outputs, and write rules from graphs.
This is review of finding outputs, finding inputs, and reading a function rule from a graph. See Section 3.4’s function-notation reminder if you need a refresher. Try the problems yourself before opening the answer key.
Check your work — Ready 1–3 answer key
-
(a) \(-13\).
(b) \(t=14\).
(c) \(21\).
(d) \(t=-14\).
-
(a) \(9\).
(b) \(x=1\).
(c) \(1\).
(d) \(g(x)=3^x\).
-
(a) \(10\).
(b) \(x=1\).
(c) \(1\).
(d) \(r(x)=7-3x\).
3. Set — Add, subtract, and compare graph outputs
Check your work — Set 4–6 answer key
Plots are not shown. Equations and points are provided where needed to check a sketch.
The packet labels two different exercises “5”; they are identified below by position.
4 — Sum graph
Rule: \(h(x)=-\frac12x+2\).
Two points: \((0,2)\) and \((4,0)\).
5 — Upper exercise: sum graph
Rule: \(s(x)=\frac35x+\frac85\).
Two points: \((-6,-2)\) and \((4,4)\).
5 — Lower exercise: parts (a)–(d)
(a) \(x=2\); intersection point: \((2,5)\).
(b) \(13\).
(c) \(2\).
(d) \(x<2\), or \((-\infty,2)\).
6 — Curve and constant function
(a) Approximately \(x<2.3\).
(b) Approximately \(0.3\).
(c) \(1.2\).
(d) \(h(x)=\frac15\).
The curve is consistent with \(r(x)=\left(\frac12\right)^x\). This is a reasonable intended rule, though the graph does not label enough exact points to determine it uniquely.
(e) Subtracting \(\frac15\) shifts the graph of \(r\) down by \(\frac15\).
Using the rule above, \(r(x)-h(x)=\left(\frac12\right)^x-\frac15\). Its y-intercept is \((0,\frac45)\), its x-intercept is approximately \((2.3,0)\), and it approaches height \(-\frac15\) to the right. Plot not shown.
Parts (a) and (b) are graph readings, so nearby estimates are reasonable.
4. Go — Discrete or continuous?
These problems review discrete and continuous models.
Several contexts do not say what the input \(x\) represents. If \(x\) counts completed deposits or games, the values form a discrete sequence. If \(x\) is elapsed time, a balance or total that changes only at deposits or games is a step function: time can take any value, but the output jumps, so the function is discontinuous. “Discrete” and “continuous” are not the only possibilities when the input is time.
Try them on your own, then open the answer key to check your work.
Check your work — Go 8–14 answer key
- If the input counts completed deposits, the balance values form a discrete sequence. If the input is elapsed time, the balance is a step function with jumps.
- Discrete: the graph shows separate points for whole numbers of bouncy balls.
- Continuous: temperature is modeled at times throughout an interval, including between whole seconds.
- If the input is the game number, the totals form a discrete sequence. If the input is elapsed time during the games, the total hits increase in jumps.
- Assuming the input is elapsed time, distance traveled is continuous throughout the trip.
- Discrete: if the input is the number of gumballs, only whole-number inputs make sense. The table lists prices for some of those counts.
- If the input counts completed monthly deposits, the balance values form a discrete sequence. If the input is elapsed time, the balance is a step function with jumps.
Additional learning resources
| Resource and direct link | What it supports | Scope / What to skip |
|---|---|---|
| Khan Academy — Adding and subtracting functions | Connects graph readings with sums and differences at the same input. | Close match to the main task and Set. |
| The Organic Chemistry Tutor — Function Operations | Worked calculations for adding and subtracting functions. | Use the addition and subtraction examples for the main task and Set. The video also covers multiplication and division, which are not needed here. |
| Paul’s Online Math Notes — Combining Functions | Notation and worked calculations for sums and differences. | Some extra material. Use the opening addition/subtraction definitions and Example 1(a)–(b). Skip products, quotients, and composition. |
| Paul’s — Combining Functions practice | Worked solutions for extra arithmetic with function values. | Some extra material. Try Problem 1(a)–(b), which evaluate differences of linear functions. Skip its product and quotient parts and later composition exercises. |
| Professor Leonard — Operations of Functions (Precalculus — College Algebra 5) | Optional extended explanation of function operations and their domains. | Some extra material. Use addition and subtraction; skip multiplication and division. Segment timestamps have not been verified. |
Teacher’s review of the source packet
Main task 5 and 7, p. 22: the linear sum’s zero is later than Aly’s emptying time. The contextual total needs either a change in rate after Aly empties or an explicit assumption that both pumps can drain a shared supply. This affects the domain, range, and emptying time. The nearby clarification distinguishes these interpretations without changing the function-addition procedure.
Coverage — Section 3.5
| Packet location | Skills supported | Where taught |
|---|---|---|
| Main 1–2 · p. 21 | Meaning of a sum; add graph heights at the same input | In class; Aly-and-Dayne table and sum graph |
| Main 3–4 · p. 22 | Add equations and connect representations | In class |
| Main 5–7 · p. 22 | Domain/range, extrema, intercepts, contextual constraints | In class; modeling clarification identifies the contextual ambiguity |
| Ready 1–3 · p. 23 | Evaluate and solve; identify linear/exponential rules from graphs | Section 2; review note and answer key |
| Set 4 and upper 5 · p. 24 | Graph the sum of two lines | In class and Section 3; answer key |
| Set lower 5(a)–(d) · p. 24 | Intersection, sums/differences, greater-than interval | Answer key |
| Set 6(a)–(e) · p. 25 | Exponential and constant rules, comparisons and difference graph | Answer key |
| Go 8–14 · pp. 25–26 | Discrete/continuous models from graphs, tables, and contexts | Review note and answer key |
Source: MVP, Secondary Mathematics One, Module 3: Features of Functions (May 2016), Section 3.5, printed pp. 21–26, and its teacher notes. The in-class lesson develops the packet’s Aly-and-Dayne example, with a redrawn sum graph. The other worked examples are original. Mathematical notation needs an internet connection to load its renderer.
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