Root functions
About this app
√x as the inverse/reflection of x² across y=x. Root index n and parameters a, c, d; domain and range highlighted.
The root as an inverse
The square root function g(x) = √x is the inverse of f(x) = x² for x ≥ 0: whatever squaring does, the root undoes. Geometrically, the graph of an inverse function is the reflection in the line y = x. More generally one studies g(x) = a·ⁿ√(x − c) + d, where a stretches the graph (and reflects it if a < 0), c shifts it right and d shifts it up.
What the app shows
Sliders for the root index n, the stretch a and the shifts c and d reshape the graph live. A switch shows the parabola x² and the mirror line y = x so the inverse relationship becomes visible. The values panel lists the function term and the parameters, and the domain and range are highlighted on the graph.
What to look for
For even indices such as 2 or 4, the root is only defined for x − c ≥ 0, so the graph starts at x = c. For odd indices such as 3 or 5, the function is defined for every real x, because negative numbers have a real cube root. Compare how domain and range change with c, d and the sign of a.
Frequently asked questions
- Why is the square root the inverse of x squared?
- For x ≥ 0, √(x²) = x and (√x)² = x, so each function undoes the other. Without the restriction x ≥ 0, x² would not be invertible, because 2 and −2 both give 4.
- What is the domain of a square root function?
- With an even index, the expression under the root must not be negative. So g(x) = √(x − c) has the domain x ≥ c. With an odd index, such as a cube root, the function is defined for all real numbers.
- How do you get the graph of an inverse function?
- Reflect the graph of the original function in the line y = x, which swaps the x- and y-coordinates of every point. This requires the function to be invertible on the interval considered, that is strictly monotonic.
- What can I adjust in the app?
- You change the root index, stretch and shifts with sliders, show the parabola and the mirror line, and see the domain and range highlighted on the graph.
Subject: School mathematics (Gymnasium)
More from School mathematics (Gymnasium)
- Tree diagram & path rules
- Binomial identities through geometry
- The number π
- Unit circle & sine curve
- Exponential growth & decay
- Ulm’s glass pyramid
- Equation balance
- Half-life & decay
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