Solving Equations with Functions

What It Means to Solve Equations

Perhaps before you’ve seen a problem like “Solve for \(x\) in the equation \(3x+1=7\).” You’ve probably also learned a few ways you could go about “solving for \(x\)” in order to get the problem done. But what does it actually mean to solve for \(x\)? What does that \(x\) mean once we have it?

Suppose you are given a function like \(f(x)=3x+1\). Recall a function is nothing more than a “machine” that takes an input (\(x\)-value) and gives you some output (\(y\)-value). So when you see an equation like \(3x+1=7\), and are asked to “find \(x\),” you are being asked “when does the function \(f(x)=3x+1\) produce an output \(y=7\)?” or rather “for what \(x\)-value does the function \(f(x)=3x+1\) produce an output of \(7\)?”

Similarly, if you are given a pair of functions, say \(f(x)=3x+1\) and \(g(x)=x^2+1\), and are asked to solve for \(x\) in the equation \(f(x)=g(x)\) (or rather \(3x+1=x^2+1\). What this is asking is “when are these two functions equal?” or “for what \(x\)-value are the \(y\)-values the same?” or, more simply: “When are the functions’ outputs the same?”

In the topics below, we cover a variety of methods for answering questions of this sort; i.e. of finding the inputs that produce specific given outputs, or for determining when two functions produce the same output.