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When we talk about absolute maximum, we mean the highest point in the range of a function on its domain. In our case, the function is \( f(x) = 1 - \sqrt{x} \). To understand this, let's explore how the graph behaves. This graph starts at the point \( (0,1) \) and continuously decreases as \( x \) increases. An important point to note here is that the absolute maximum is the greatest value that the function achieves. For this specific function, the absolute maximum is observed right at the point of origin on the graph, \( x = 0 \), where \( f(0) = 1 \). Since there is no higher value that \( f(x) \) can take on within its valid domain of \( x \geq 0 \), \( f(0) = 1 \) is the absolute maximum.

• The topmost point in the function's range.
• Occurs at the beginning of the function's domain.
• The highest value the function can produce.

Local minimum and maximum refer to the points in the graph where a function changes direction from increasing to decreasing or decreasing to increasing, respectively. However, it's important to understand that not all functions have these points. In the function \( f(x) = 1 - \sqrt{x} \), the shape of the graph shows that it is strictly decreasing, meaning it does not turn back upwards or change its direction at any point. This continual decrease implies there are no local maxima or minima within the domain.

• Local extrema are points where increments stop and decrements start, or vice versa.
• If the function only decreases or increases, local extrema won't appear.
• In this case, since \( f(x) \) monotonically decreases, there are neither local minimums nor maximums.

Function transformations involve changes that modify the graph of a basic function in various ways without altering its overall shape or direction. With \( f(x) = 1 - \sqrt{x} \), we see a combination of such transformations from the basic \( \sqrt{x} \), or square root function. There are two main transformations applied here:- **Vertical Reflection**: The negative sign before the square root, \( -\sqrt{x} \), reflects the graph across the x-axis. This turns the usual upwards opening root graph downwards.- **Vertical Shift**: The addition of 1, shown as \( 1 - \sqrt{x} \), moves the entire graph upwards by 1 unit. To visualize these transformations:

• Start with the basic \( \sqrt{x} \) graph, originating at \( (0,0) \).
• Reflect it vertically to point downwards, resulting in \( -\sqrt{x} \).
• Shift every point of this graph upwards by one unit.

These transformations help us understand key characteristics and behavior of \( f(x) \), aiding in easier graph sketching and extrema identification.