Sketch the graph of the function using the approach...

Key Concepts

Function Transformations and Composition

Graphing complex functions can be simplified by recognizing them as transformations of simpler, basic functions. In this context, understanding that the function comprises a linear term x and a shifted fractional power term (x - 1)^(1/5) allows one to interpret the graph in terms of horizontal translations and scaling, thereby facilitating an effective sketch of the overall behavior.

End Behavior

End behavior describes the behavior of a function as x approaches positive or negative infinity. Analyzing the dominant terms of the function helps predict whether the graph will rise, fall, or level off at the extremes, which is essential for a comprehensive sketch.

Critical Points and Derivative Analysis

Critical points are values of x where the derivative of the function is zero or undefined, and they help identify regions of local maxima, minima, or inflection. With functions involving fractional exponents, careful differentiation is required since the derivative may behave non-trivially near points where the inside of the radical equals zero or changes sign.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. Determining the x-intercepts involves setting the function equal to zero and solving for x, while the y-intercept is found by evaluating the function at x = 0. These points serve as anchor points for sketching the graph.

Domain

The domain of a function is the set of all input values for which the function is defined. When dealing with fractional (or radical) exponents, it is important to note whether the radical is odd or even. Odd roots, such as the fifth root in this example, allow the function to be defined for all real numbers, whereas even roots may impose restrictions.

Transcript

00:01 We'll ask to plot x times the fifth root of x minus 1.

00:05 So x times x minus 1 to the 1 fifth power.

00:12 Obviously when x is 0, our function is 0, so we have 0.

00:20 Now as x gets very, as x gets to be a very negative number, this is going to be negative, and then this is going to be negative.

00:30 So we're going to go off to positive infinity there.

00:33 When x gets very large, this is positive.

00:37 This is positive, so we're going to go off to positive infinity on the right side, as x gets very, is x goes to infinity.

00:46 Now, we can take some derivatives here, and so the first derivative, we get 6x minus 5 all over 5 times x -1 to the 4th, 5th power.

00:58 So again, just using the product rule here, and then some simplifications, we get this.

01:04 And so what do we see here? well, when x is equal to 5 sixth, that we have a vertical horizontal tangent, so that's this point here.

01:15 So 56 is 0 .825.

01:20 So we get this.

01:22 Plugging that back into here, that's minus 0 .58.

01:28 So that point right there is our, we have a horizontal tangent.

01:33 Now clearly we have a vertical tangent and x equals 1.

01:37 So when x equals 1, this thing is 0 again.

01:43 So 1 -0, and we know we have a horizontal tangent there, or a vertical tangent and a horizontal tangent here.

01:50 Now, take another derivative when we get 2 times 3x minus 5 all over 25 x minus 1 to 9 5th.

01:59 Again, after some simplification, you know, using the quotient rule, and then a bunch of simplification, we get this.

02:08 And so that's 0 when x equals 5 thirds, or 1 and 2 thirds, but 1 .67, plug that into here, and we get 1 .5 .4.

02:18 So that's another point.

02:19 That's an inflection point where the curvature is zero.

02:24 And again, we can see that the curvature is infinite at x equals 1...

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