Question

$f(x) = (x + 1)^2$ for $x \ge 1$. $g(x)$ is a function whose graph is the reflection of the graph of $f(x)$ in the line $y = x$, then $g(x)$ is

• A. $-\sqrt{x} - 1$
• B. $\sqrt{x} + 1$
• C. $\sqrt{x} - 1$
• D. $\sqrt{-x} - 1$

Hint:

Always remember: "Reflection across $y = x$" is the code phrase for "Find the inverse function". The domain restriction ($x \ge 1$) is a critical hint that you must carefully consider whether to take the positive or negative branch when square rooting.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Reflection of a graph in the line $y = x$ gives the inverse function.
Hence, $g(x) = f^{-1}(x)$.
Step 2: Write the Function
\[ y = (x + 1)^2, x \ge 1 \]
Step 3: Interchange $x$ and $y$
\[ x = (y + 1)^2 \]
Step 4: Solve for $y$
\[ y + 1 = \pm \sqrt{x} \] \[ y = -1 \pm \sqrt{x} \]
Step 5: Choose Correct Branch
Since $x \ge 1$, the range of $f(x)$ is $y \ge 4$.
Thus, for inverse function: \[ y \ge 1 \] So, take positive root: \[ y = \sqrt{x} - 1 \]
Step 6: Final Answer
\[ \boxed{g(x) = \sqrt{x} - 1} \]