Sketch the graph of f by hand and use your sketch to find...

Sketch the graph of $ f $ by hand and use your sketch to find the absolute and local maximum and minimum values of $ f $. (Use the graphs and transformations of Section 1.2 and 1.3).

$ f(x) = 1 - \sqrt{x} $

Key Concepts

Properties of the Square Root Function

This concept covers the fundamental characteristics of the square root function, including its domain, range, and general shape. Recognizing that the square root function is only defined for non-negative inputs and that its basic form increases slowly helps in understanding how additional transformations, such as reflections or shifts, will affect its graph.

Graphing Transformed Functions

This concept involves taking a basic function and applying various modifications such as shifts, reflections, stretches, or compressions to obtain the graph of a more complex expression. It requires understanding how the parent function's graph is altered by operations inside or outside the function, helping to quickly sketch the new graph by relating back to the original simple curve.

Absolute and Local Extrema

This concept pertains to determining the highest or lowest points on a function’s graph. Absolute (or global) extrema refer to the overall maximum or minimum values on the entire domain of the function, while local (or relative) extrema are peaks or valleys within a specific interval. Analyzing these points is key in calculus for optimization and understanding the behavior of functions.

Transcript

00:02 We are going to sketch the graph of the function 1 minus square root of x, and we'll use that graph to find the absolute and local maximum and minimum values of the function.

00:18 So there is no domain specified here, so we are going to use the domain implicit in the formula, that is, in this case, we have a square root of x, and that needs, or is only defined for x, greater than or equal to zero.

00:36 So we are considering f of x equal 1 minus square root of x for x on the interval 0, including 0, and plus infinity.

00:51 That is for the non -negative number x.

00:55 Then we know that the square root of x is inverse of x squared for the positive numbers.

01:07 That is, we're going to do some.

01:09 Transformation to arrive to the graph of square root of x.

01:15 So we have this, let's say the parabola x squared something like this.

01:22 It has a negative part but we don't, that is a part which corresponds to the images of the negative numbers, which is just a reflect, a reflection of this respect to the y -axis.

01:38 But we know that we have to consider only x -positive so this is the part of the parabola x square we got to consider.

01:47 So here we have, let's say, y equal x squared for x greater than equal 0.

01:57 And we know that to find that the inverse of that, we get to draw the identity function that is the line, y equals what, y equals x.

02:11 Let's say that.

02:13 And with that we made a reflection.

02:16 Respect to that line and we get the inverse of the function.

02:19 So we are going to get something just a little bit like this.

02:28 Let's say something like not so bad like this, but to make a little bit idea of the reflection.

02:39 Okay.

02:41 And it's endless to the right, as was the parabola...