Finding the Input for a Given Output
Finding the input for a given output means setting the function's formula equal to that output and solving the resulting equation, which can have one, two, or no solutions depending on the function.
Evaluating a function means plugging in x and getting a number out. This topic runs that process backwards: you're given the output and asked which input (or inputs) produced it. The move is always the same. Replace f(x) with the target number, and you've turned a function into an equation. From there you solve for x using whatever tools the formula demands: isolating a variable for linear expressions, squaring both sides for a square root, factoring or the quadratic formula for a quadratic, or logarithms for an exponential.
Why can there be more than one answer? Because many functions aren't one-to-one. A line hits every output exactly once, so running it backwards gives exactly one input. But a parabola doubles back on itself: except at its vertex, every output to one side is matched by a mirror-image output on the other side, so two different inputs can give the same result. An exponential function, by contrast, is one-to-one, so solving 500 times 2t = 8000 gives a single value of t.
This idea connects directly to the shape of the graph next to it. The number of inputs for a given output is just the number of times a horizontal line at that height crosses the graph. For a parabola, that count is two above the vertex, one at the vertex, and zero below it (for an upward-opening one). Keeping that picture in mind prevents a common error: assuming an equation must have two solutions just because it came from a quadratic.
Key terms
- One-to-one function
- A function where every output comes from exactly one input, so reversing it never gives more than one answer.
- Vertex
- The highest or lowest point of a parabola, where the output is produced by only one input.
- Extraneous solution
- A value that solves a transformed equation but fails in the original, often introduced by squaring both sides.
- Inverse operation
- An operation that undoes another, such as dividing to undo multiplication, used to isolate x.
Watch it explained
- How it worksHow does setting the formula equal to a target output turn an evaluation into an equation to solve?
- Worked exampleFind every x with f(x) = 12 when f(x) = x2 + 4x.
- Worked exampleFind t when 500 times 2 to the t equals 8000.
- Worked exampleFind x when the square root of (x-1) equals 6, and then when 12/(x+2) equals 3.
- Common mistakeWhy doesn't every output of a quadratic come from exactly two different inputs?
Check yourself
End of topic.