91Ó°ÊÓ

When sketching graphs, it's like painting a picture of an equation. In our case, we have a downward-opening parabola described by the function \( f(x) = 16 - x^2 \). This indicates that the curve opens downward, and the highest point, or vertex, is at \((0, 16)\).
Imagine drawing a smiley face, but with only half a curve because of the condition \( x \geq 0 \). This means you only draw the right half, starting from the vertex straight down. The more detail, the more informative your graph.
Sketching the inverse function \( f^{-1} \) involves a reflection over the line \( y = x \). This can be thought of as flipping the graph so that every point \((a, b)\) becomes \((b, a)\). It’s like looking into a mirror and swapping your x and y coordinates.
So, by accurately sketching both the function and its inverse, you gain a complete visual understanding of their relationship.

The domain and range of a function are essential to define its scope and reach. For our parabolic function, \( f(x) = 16 - x^2 \), we have a specific domain: \( x \geq 0 \). This tells us that we're only considering the right half of the parabola.
The range for this function is \(0 \leq y \leq 16\), as the parabola peaks at 16 and extends downward. It's the full set of potential outputs or y-values the function can achieve.
Now, consider the inverse function \( f^{-1}(x) \). The domain of the original becomes the range of the inverse and vice versa. Thus, \( f^{-1}(x) = \sqrt{16 - x} \) has a domain \(0 \leq x \leq 16\) and range \(y \geq 0\). Understanding and defining these domains and ranges is crucial when working with inverse functions.

• Original Function: \( x \geq 0 \) (Domain), \(0 \leq y \leq 16\) (Range)
• Inverse Function: \(0 \leq x \leq 16\) (Domain), \(y \geq 0\) (Range)

Parabolas are fascinating structures in algebra and geometry. At their essence, parabolas form symmetrical, U-shaped curves.
Our equation \( f(x) = 16 - x^2 \) defines a downward-opening parabola. The negative coefficient before \(x^2\) signals this downward opening, like a frown.
Each parabola has a key feature: the vertex. This is the point where the parabola changes direction. For our equation, the vertex is at \((0, 16)\), since without any additions to \(x\), its y-intercept is also its vertex.
Graphing parabolas requires observing their properties:

• Axis of symmetry: This is the vertical line that divides the parabola into mirror images. Here, it’s the y-axis or \(x = 0\).
• Vertex: The highest or lowest point, depending on the parabola’s direction. In our case, it’s the highest point.
• Direction: Determined by the sign of the \(x^2\) term. Negative means down.

Understanding these core elements makes working with parabolas intuitive and manageable.