Evaluating a Linear Function and Reversing It
f(x) is a machine: you feed in a number for x, the rule acts on that exact number, and one output comes out. Running the machine backwards means undoing the operations in reverse order to recover the input.
f(x) is just a naming system for a rule. Wherever you see x in the rule, you replace it with whatever is inside the parentheses after f. So f(3) does not mean multiply f by 3 — f is not a number, it's the name of the machine. It means take the rule, put 3 in for x, and compute. For f(x) = 7 - 2x, f(4) means replace every x with 4: 7 - 2(4) = 7 - 8 = -1. The same logic handles an expression as input: f(x + 2) replaces x with the whole chunk x + 2, not just part of it, which is why f(x + 2) is not f(x) + 2. For f(x) = 3x - 1, f(x + 2) = 3(x + 2) - 1 = 3x + 5, while f(x) + 2 = 3x + 1. Different rules, different answers.
Reversing the machine means you know the output and want the input. Set f(x) equal to the given value and solve for x, undoing operations in the opposite order from how they were applied. Since the rule multiplies x then subtracts, undoing it means adding back first, then dividing. For f(x) = 7 - 2x, finding x when f(x) = 15 means solving 7 - 2x = 15, giving x = -4.
This skill is the backbone of reading function graphs and tables on the exam: evaluating is reading a y-value off an x-value, and reversing is reading an x-value off a y-value. It also sets up systems of equations and transformations of functions, where you compose or shift rules instead of just plugging in one number.
Key terms
- function
- A rule that takes each input and assigns it exactly one output.
- input
- The value placed into a function in place of x.
- output
- The result of running an input through a function's rule, often written f(x).
- evaluate
- To substitute a specific value for x and compute the resulting output.
- inverse operation
- An operation that undoes another, used in reverse order to recover an input from an output.
Watch it explained
- How it worksHow does a function machine take an input, run it through a rule, and hand back exactly one output?
- Worked exampleFind f(4) when f(x) = 7 - 2x, and then find the x that makes f(x) equal 15.
- Common mistakeWhy isn't f(x + 2) the same thing as f(x) + 2 for the function f(x) = 3x - 1?
- Going furtherWhat changes when you run the machine backwards to recover the input that produced a given output?
Check yourself
End of topic.