Inverse of a Square Root Function

Step‑by‑step algebra solver for square root functions of the form \( f(x) = a\sqrt{bx + c} + d \). Generate random functions, see detailed algebraic steps with actual numbers, determine the range, and visualize the function and its inverse reflected across \( y = x \).

📐 Interactive Calculator • Purple Theme • Square Root Functions

✨ A square root function \( f(x) = a\sqrt{bx + c} + d \) (with \( a \neq 0 \), \( b \neq 0 \)) has an inverse that is a quadratic function. To find the inverse: swap \(x\) and \(y\), solve for \(y\), and determine the domain restriction from the range of \(f\). Below, generate random examples, see detailed steps, and visualize symmetry.
\( f(x) = 2\sqrt{x - 1} + 3 \)

📖 Step-by-Step Solution

STEP 1: Replace \(f(x)\) by \(y\)
—
STEP 2: Solve for \(x\)
—
STEP 3: Interchange \(x\) and \(y\)
—
STEP 4: Find the range of \(f\) (domain of \(f^{-1}\))
—
STEP 5: Write the inverse function with its domain
—
📊 Graphical meaning
\(f\) (green) and \(f^{-1}\) (blue) are symmetric about \(y = x\) (red).

Adjust axis scale to explore reflection property. Note: Square root function domain is where \(bx + c \ge 0\).