Comparing Two Functions, and Working Backwards

In this lesson you will: Translate a graph, a table, or a description into y=mx+b so two functions can be compared; Compare rates of change and values at a chosen input across different representations; Find the input that produces a given output.

Translate first, compare second

Ohio likes to hand you two functions in two different costumes: one as a graph, one as an equation, one as a table, one as a paragraph. The comparison itself is easy. The trap is comparing before translating.

Put both into the same form and the question answers itself. The rate of change and the value at input zero are the two features worth comparing, and every representation will give you both.

Where two models meet

Once both are in the same form, setting them equal locates the one input where they agree — which is Chapter 3's work, now with a purpose.

Take f(x)=4x+9 against a table giving g as 3,13,23,33 at inputs 0,2,4,6. The table's rate is 10/2=5 and its value at zero is 3, so g(x)=5x+3. Then g rises faster (5>4) while f starts higher (9>3). Setting 4x+9=5x+3 gives x=6, where both equal 33.

Working backwards: solving f(x) = k

Every question so far has run the function forwards. Now the output is known and the input is not: when does the candle reach 7 centimeters, how many bowls give $110 of profit, what input produces 30?

From an equation, replace f(x) with the known output and solve. From a table, scan the output column and read across. From a graph, draw the horizontal line at that height and read the input where it crosses. Three representations, one idea.

Common mistake — Answering f(x) = k by computing f(k)

Asked to find the input for which f(x)=30, with f(x)=7x-12, a student computes f(30)=7(30)-12=198 and writes 198.

Nearly every exercise up to this point handed over an input and asked for an output, so the fingers move before the sentence is read. The number 30 is right there; substituting it feels like the obvious use for it.

But 30 arrived as an output. It belongs on the same side as f(x), not inside the parentheses. Solving 7x-12=30 gives 7x=42 and x=6.

The two answers are not close, and they are not even the same kind of thing: 198 is an output and 6 is an input.

How to catch it: Substitute your answer and look at what comes out. f(6)=42-12=30 is the number you were given, so 6 is right; f(198) is nowhere near 30.