Introduction — What Is the Square Root of a Function?. Understand the big idea: replacing x with f(x) inside the square root creates a new function y = √(f(x)). Learn why the domain is restricted to where f(x) ≥ 0, how to build the graph using tables (keeping x-values, taking square roots of y-values), and discover invariant points where both graphs meet at y = 0 and y = 1.
Example 1 — Working with y = √(4 − 2x). Master a complete example with a decreasing linear function. Find the domain restriction (x ≤ 2), build a table of values, identify both invariant points (2,0) and (3/2,1), and learn to factor the radicand to determine domain algebraically. Includes a comprehensive summary of key facts about y = √(f(x)).
Reading Graphs and Finding Equations. Learn to work backwards from graphs. Find g(x) from a line graph and write √(g(x)) in standard form. Use invariant points to determine f(x) when given √(f(x)). Master the prediction table that tells you where √(f(x)) sits relative to f(x) based on the value of f(x).
Mapping Points & Diploma Practice. Master the complete mapping formula for transformations of the form y = a√(b(x − h)) + k. Learn to find image points P' from original points P using x' = x + h and y' = a√(by) + k. Practice diploma-style problems including finding endpoints from radical equations in factored form.
Inside the lesson — a free preview
The Big Idea
You already know how to graph y = √(x) (the parent function). Now the question becomes: what happens when the thing inside the square root is not just x, but an entire function — like x - 2, or 4 - 2x?
When you replace x with f(x) inside the radical, you get a brand new function:
The Square Root of a Function. y = √(f(x))
This means: take any function f(x), and take the square root of its output.
What Does y = √(f(x)) Mean?
It's a simple idea — but it has some important consequences.
⚠️ The Most Important Rule:. y = √(f(x)) is only defined where f(x) ≥ 0
You cannot take the square root of a negative number, so the domain of y = √(f(x)) is restricted to only the x-values where f(x) is zero or positive.
Worked example: Worked Example: f(x) = x - 2
Given f(x) = x - 2, let's build the square root function y = √(x - 2).
📊 What the Table Shows. • For x < 2, the f(x) values are negative — so y = √(x-2) is undefined there
• The square root function only "switches on" once f(x) reaches 0 at x = 2
• After that, as f(x) increases, √(f(x)) increases more slowly (because square roots compress larger values)
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