6.4 Comparing and Combining Functions Across Representations

6.4 Comparing and Combining Functions Across Representations

Every F-IF.9 item has the same shape. You are handed two functions in two different formats and asked which one has the greater something. The format asymmetry is the difficulty; the underlying comparison is usually easy once each property is extracted.

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1. The Extraction Table

Do not convert everything into an equation. Pull the one property you need directly from whatever form you were given.

Fingerprinting a Table

For equally spaced inputs:

• Constant first differences $\Rightarrow$ linear.
• Constant second differences $\Rightarrow$ quadratic.
• Constant ratio between consecutive outputs $\Rightarrow$ exponential.
• Neither pattern, but a curve that flattens as $x$ grows and is defined only for $x \geq$ some value $\Rightarrow$ suspect a square root; defined everywhere with an inflection $\Rightarrow$ suspect a cube root.

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2. Worked Comparison 1: Equation Versus Table

Problem: Function $f$ is defined by $f(x) = 2x^2 - 8x + 3$. Function $g$ is given by the table below. Which function has the greater minimum value, and by how much?

• Confirm the family of $g$. First differences: $-6, -2, 2, 6$. Second differences: $4, 4, 4$. Constant second differences confirm a quadratic.
• Minimum of $g$. The table is symmetric about $x = 2$, where the output $-2$ is the smallest listed value and the outputs rise identically on both sides. The minimum value of $g$ is $-2$.
• Minimum of $f$. For a quadratic in standard form the vertex sits at $x = -\dfrac{b}{2a} = -\dfrac{-8}{2(2)} = 2$. Then $f(2) = 2(4) - 8(2) + 3 = 8 - 16 + 3 = -5$. Because $a = 2 > 0$, this is a minimum.
• Compare. $-2 > -5$, so $g$ has the greater minimum, by 3 units.

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3. Worked Comparison 2: Verbal Versus Graph, Using Average Rate of Change

Problem: A biologist describes colony $A$ verbally: it starts at 40 cells and triples every hour. Colony $B$ is shown on a graph that passes through $(0, 400)$, $(1, 600)$, $(2, 800)$, and $(3, 1000)$. Over the interval from $t = 0$ to $t = 3$ hours, which colony has the greater average rate of change?

• Model $A$ from the description. Tripling per hour is exponential: $A(t) = 40(3)^t$. Then $A(0) = 40$ and $A(3) = 40(27) = 1080$. $$\text{AROC}_A = \frac{1080 - 40}{3 - 0} = \frac{1040}{3} \approx 346.7 \text{ cells per hour}$$
• Read $B$ from the graph. The listed points rise by a constant 200 per hour, so $B$ is linear with $B(0) = 400$ and $B(3) = 1000$. $$\text{AROC}_B = \frac{1000 - 400}{3 - 0} = \frac{600}{3} = 200 \text{ cells per hour}$$
• Compare. $346.7 > 200$, so colony $A$ has the greater average rate of change over $[0, 3]$.
• Interpret honestly. Colony $B$ is far larger for the first two hours - $A(1) = 120$ against $B(1) = 600$ - and only overtakes $B$ between $t = 2$ ($A = 360$, $B = 800$) and $t = 3$ ($A = 1080$, $B = 1000$). "Greater average rate of change on an interval" is not the same claim as "larger throughout."

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4. Combining Functions with Arithmetic Operations (AII-F.BF.1b)

Standard AII-F.BF.1b asks you to build a new function from two standard types using $+$, $-$, $\times$, or $\div$. The workhorse example in context is profit.

$$P(x) = R(x) - C(x)$$

Worked Combination

Problem: A small manufacturer's weekly revenue in dollars from selling $x$ units is $R(x) = 120x - 2x^2$, and weekly cost is $C(x) = 300 + 20x$. Write the profit function, determine the profit at 10 units, and find the production level that maximizes profit.

• Combine. $$P(x) = (120x - 2x^2) - (300 + 20x) = -2x^2 + 100x - 300$$ Distribute the subtraction across both terms of $C(x)$ - dropping the sign on $+20x$ is the single most common error here.
• Evaluate at 10 units. $$P(10) = -2(100) + 100(10) - 300 = -200 + 1000 - 300 = 500$$ Weekly profit at 10 units is $\text{\textdollar}500$.
• Maximize. The profit function is a downward-opening parabola, so its maximum is at the vertex: $$x = -\frac{b}{2a} = -\frac{100}{2(-2)} = 25 \text{ units}$$ $$P(25) = -2(625) + 100(25) - 300 = -1250 + 2500 - 300 = 950$$ Maximum weekly profit is $\text{\textdollar}950$ at 25 units.

Rules a Combined Function Inherits

For example, if $f(x) = \sqrt{x - 1}$ and $g(x) = x - 4$, then $(f + g)(x) = \sqrt{x - 1} + x - 4$ has domain $[1, \infty)$, while $\left(\frac{f}{g}\right)(x)$ has domain $[1, 4) \cup (4, \infty)$.