Image and Preimage

The Two Directions

A function has arrows going from inputs to outputs.

You can ask two questions:

Forward: Where does this input go?

Backward: What input(s) produced this output?

Image: Following Arrows Forward

Pick an input. Follow its arrow. Where does it land?

That’s the image.

Image of a Set

You can also ask: where does a group of inputs land?

Collect all their outputs. That’s the image of the set.

Example:

$f(x) = x^2$

Notice the last one: two inputs, but only one output (both square to 4).

Preimage: Tracing Arrows Backward

Pick an output. Trace all arrows pointing to it. Where did they come from?

Those inputs are the preimage.

Preimage Can Be Empty, One, or Many

This is the key insight:

Solving Equations = Finding Preimages

Here’s WHY this matters:

Solving an equation IS finding a preimage.

$\text{Solve } x^2 = 9$

You’re asking: what inputs produce output 9?

$\text{Preimage of 9} = \{-3, 3\}$

You’ve been finding preimages your whole life — just didn’t call it that.

More Examples

$f(x) = x^2$

The preimage tells you everything about solvability.

Preimage Notation Looks Like Inverse

$f^{-1}(\{9\}) = \{-3, 3\}$

Don’t be confused! This is NOT the inverse function.

Preimage always exists. Inverse might not.

Key Facts

• Image = where inputs land (forward)
• Preimage = where outputs came from (backward)
• Preimage is always a set (empty, one, or many elements)
• Solving = finding preimages
• Image of entire domain = range