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sketch the inverse of the following function

To sketch the inverse of the given function, let's first identify the function depicted in the graph.

The graph is a parabola with the following characteristics:
- It opens upwards.
- The vertex is at the point \((0, 2)\).
- The axis of symmetry is the y-axis.

Given the vertex and the general shape, the equation of the parabola can be written in vertex form as:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. Here, the vertex \((0, 2)\) gives us:
\[ y = ax^2 + 2 \]

To find \(a\), we need another point on the graph. By inspecting the graph, a notable point is \((1, 3)\). Substituting this point into the equation:
\[ 3 = a(1)^2 + 2 \]
\[ 3 = a + 2 \]
\[ a = 1 \]

So, the equation of the parabola is:
\[ y = x^2 + 2 \]

To find the inverse of this function, we switch \(x\) and \(y\) and solve for \(y\):
\[ x = y^2 + 2 \]
\[ x - 2 = y^2 \]
\[ y = \pm\sqrt{x - 2} \]

This represents two functions:
\[ y = \sqrt{x - 2} \]
\[ y = -\sqrt{x - 2} \]

The inverse of the given function is not a single function but two separate branches. To graph these, we plot:
\[ y = \sqrt{x - 2} \] and \[ y = -\sqrt{x - 2} \]

### Sketch of the Inverse Functions:
1. Plot the line \(y = \sqrt{x - 2}\):
   - Starts at \((2, 0)\) and opens to the right.
   - Only defined for \(x \geq 2\).

2. Plot the line \(y = -\sqrt{x - 2}\):
   - Starts at \((2, 0)\) and opens to the right.
   - Only defined for \(x \geq 2\).

Here is a description of how the sketch should look:
- The two branches will mirror each other across the x-axis, starting from the point \((2, 0)\).
- The upper branch will rise as \(x\) increases, and the lower branch will fall as \(x\) increases.

If you have any questions or need further details, please let me know.

### Questions you might want to ask next:
1. How do I determine the domain and range of an inverse function?
2. What are the steps to find the inverse of a more complex function?
3. Can you explain why some functions do not have inverses?
4. How can I verify if a function and its inverse are correct?
5. What are the real-world applications of inverse functions?

### Tip:
To verify if two functions are inverses of each other, compose them (i.e., \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \)) and check if both compositions return the input \(x\).

Learn how to sketch the inverse of a quadratic function with a detailed step-by-step guide. This problem covers key concepts in functions and quadratic equations, including the vertex form of a quadratic equation. Suitable for students in Grades 10-12.

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