Reading f(x) in Context — What f(3) = 20 Means

In this lesson you will: Translate a function statement into a sentence carrying the right units; Tell the difference between \"find f(8)\" and \"solve f(x)=8\"; Read inputs and outputs off a graph that models a real situation.

Every statement is two numbers wearing different units

The statement f(3)=20 says nothing at all until you know what f measures. Suppose f(t) is the number of gallons in a tank t minutes after the pump starts. Then f(3)=20 means: three minutes in, there were twenty gallons.

The number inside the parentheses always wears the input's units. The number after the equals sign always wears the output's units. Write both units down before you write the sentence and the interpretation writes itself.

Evaluate, or solve? Decide before you compute

\"What does 6 gigabytes cost?\" gives you the input and wants the output. Evaluate.

\"How much data gave a bill of $79?\" gives you the output and wants the input. Solve.

These two get confused constantly, and the cost is total — the arithmetic is fine and the answer is an answer to a different question. The giveaway is the units of the number the question hands you.

Reading a graph that models something real

On a graph the same two rules apply, with horizontal position as input and height as output. To find h(3), go to 3 on the horizontal axis, go up to the curve, read across. To solve h(t)=28, start at 28 on the vertical axis, go across to the curve, and read down — and be ready for more than one answer.

That last warning matters. A ball thrown upward passes every height below its peak twice, once going up and once coming down. Two inputs sharing an output is the ordinary behavior from Lesson 7.1, and it is exactly what you see here.

Common mistake — Reading f(3) = 20 backwards

Told that C(4)=86, where C(n) is the cost in dollars of n shirts, a student writes \"it costs $4 for 86 shirts.\"

The two numbers are both there, so the sentence feels complete, and nothing in the arithmetic objects. On a multiple-choice item the reversed reading is almost always one of the options.

The input is whatever sits inside the parentheses. Always. Here that is 4, so four shirts cost eighty-six dollars.

The same reversal turns up in the other direction too: asked to solve C(n)=86, a student computes C(86) instead and produces a number in the thousands without blinking.

How to catch it: Attach units to both numbers before you write the sentence. C(4)=86 becomes \"4 shirts, $86\" — and \"$4 for 86 shirts\" is then visibly the wrong pairing. If the units in your sentence are on the wrong numbers, you have read it backwards.