Where f(x) Meets g(x)

In this lesson you will: The solutions of f(x)=g(x) are the x-coordinates of the points where y=f(x) and y=g(x) intersect; Use a graph or a table to locate a solution, and narrow it by successive approximation; Tell the difference between the solution of f(x)=g(x) and the intersection point itself.

An old idea in new clothes

In Chapter 3 you solved 4x-7=21 and saw the picture beside it: the line y=4x-7 reaching a height of 21. Solving an equation was finding where two expressions agree.

Nothing about that changes when both sides are interesting. Call the left side f(x) and the right side g(x). The equation f(x)=g(x) asks for the inputs where the two functions produce the same output, and on a graph those are exactly the inputs where the curves meet.

That is the whole of A.REI.11, and it applies whether f and g are lines, absolute values, or growth curves.

When the crossing is not a nice number

Plenty of equations have solutions that no amount of rearranging will produce exactly at this level. You can still find them to any accuracy you like by squeezing.

Build a table of f(x) and g(x) and watch which one is larger. Wherever the larger one changes, a crossing has happened in between. Then split that interval and test again.

Take f(x)=2^x and g(x)=3x. At x=3, f is 8 and g is 9, so f is behind. At x=4, f is 16 and g is 12, so f is ahead. Somewhere between 3 and 4 they were level.

The question decides what to write down

\"Solve f(x)=g(x)\" wants the x-value or values. \"Find the point of intersection\" wants an ordered pair. \"For what value does the cost of the two plans agree, and what is that cost?\" wants both, in words.

Read the last line of the question before you write anything. The arithmetic is the same; only the form of the answer changes.

Common mistake — Reporting the height instead of the input

Two graphs meet at (5,9). Asked to solve f(x)=g(x), a student writes 9.

The 9 is the visible, memorable number — it is where the crossing sits on the page, and it is the value both functions produce. It feels like the answer.

But f(x)=g(x) is an equation in x. Its solutions are inputs. At x=5 the two functions agree, so 5 is the solution; 9 is what they agree on.

How to catch it: Name the variable in the equation you are solving. f(x)=g(x) has x in it, so the answer is a value of x. If you wrote down the second coordinate of the crossing point, you answered a different question.