Answer
To find the inverse of \( f(x) = x^{2} \), restrict the domain to \( x \geq 0 \), resulting in \( f^{-1}(y) = \sqrt{y} \). Graphically, this inverse is the reflection of the original parabola across the line \( y = x \).
Solution
To find the inverse of the function \( f(x) = x^{2} \) and understand the implications for its graph, follow these steps:
### 1. **Understand the Function \( f(x) = x^{2} \)**
The function \( f(x) = x^{2} \) is a quadratic function, which graphs as a parabola opening upwards with its vertex at the origin \((0, 0)\). However, this function is **not one-to-one** over its entire domain (all real numbers) because each positive \( y \) value corresponds to two different \( x \) values (one positive and one negative).
### 2. **Define the Inverse Function Conceptually**
An inverse function essentially "reverses" the original function. For \( f(x) = x^{2} \), the inverse would take a \( y \) value and return the corresponding \( x \) value such that \( y = x^{2} \).
### 3. **Solve for \( x \) in Terms of \( y \)**
Start by expressing \( f(x) \) as:
\[
y = x^{2}
\]
Solve for \( x \):
\[
x = \pm \sqrt{y}
\]
This equation suggests two possible inverses:
\[
x = \sqrt{y} \quad \text{and} \quad x = -\sqrt{y}
\]
However, a function must have only **one output** for each input, so we cannot have both \( \sqrt{y} \) and \( -\sqrt{y} \) simultaneously in the inverse.
### 4. **Restrict the Domain to Make \( f(x) \) One-to-One**
To define an inverse function, we must restrict the domain of \( f(x) \) so that it becomes one-to-one (each \( y \) corresponds to exactly one \( x \)). Typically, we consider two cases:
- **Case 1: \( x \geq 0 \) (Right Branch)**
\[
f(x) = x^{2} \quad \text{with} \quad x \geq 0
\]
The inverse function is:
\[
f^{-1}(y) = \sqrt{y}
\]
Defined for \( y \geq 0 \).
- **Case 2: \( x \leq 0 \) (Left Branch)**
\[
f(x) = x^{2} \quad \text{with} \quad x \leq 0
\]
The inverse function is:
\[
f^{-1}(y) = -\sqrt{y}
\]
Defined for \( y \geq 0 \).
For the inverse to be a function, **only one branch** can be used. Commonly, the **right branch (\( x \geq 0 \))** is chosen, resulting in:
\[
f^{-1}(y) = \sqrt{y} \quad \text{for} \quad y \geq 0
\]
### 5. **Implications for the Graph**
- **Inverse Function Graph**: The graph of the inverse function \( f^{-1}(y) = \sqrt{y} \) is a **half-parabola** that is the **reflection** of the original parabola \( f(x) = x^{2} \) across the **line \( y = x \)**.
- **Graphical Representation**:
- The original function \( f(x) = x^{2} \) is symmetric about the \( y \)-axis.
- The inverse function \( f^{-1}(y) = \sqrt{y} \) is defined only for \( y \geq 0 \) and lies entirely in the first quadrant.
- Plotting both the function and its inverse on the same axes, you will see that each point \( (a, b) \) on \( f(x) \) corresponds to a point \( (b, a) \) on \( f^{-1}(y) \).
- **Example Visualization**:
\[
\begin{align*}
&\text{Original Function:} \quad y = x^{2} \\
&\text{Inverse Function:} \quad y = \sqrt{x} \\
&\text{Reflection Line:} \quad y = x
\end{align*}
\]
The inverse function \( y = \sqrt{x} \) will appear as the mirror image of \( y = x^{2} \) across the line \( y = x \).
### 6. **Conclusion**
Because \( f(x) = x^{2} \) is not one-to-one over all real numbers, its inverse is not a function unless we restrict its domain. By limiting the domain to \( x \geq 0 \), we obtain the inverse function \( f^{-1}(y) = \sqrt{y} \), and graphically, the inverse is the reflection of the original parabola across the line \( y = x \).
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