Graph ƒ₁ and ƒ₂ together. Then describe how applying the absol...

Question

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)

__ ___

√ x √ |x|

Accepted Answer

Hi everyone. Let's take a look at this practice problem. This problem says consider the functions F1 of X, which is equal to the square root of the quantity of 1/2 multiplied by X in quantity, and F2 of X, which is equal to the square root of the absolute value of the quantity of 1/2 multiplied by X in quantity. Graph them together and describe how applying the absolute value function in F2 modifies the graph of F1. Below the problem we're given an empty graph on which to plot our functions. So, the first part of this problem wants us to create our graph. So, to do this, we need to find points for our two functions, F1 and F2 to plot on our graph. So, we're going to start off by looking at F1. So if we look at F1 of X, which is equal to the square root of the quantity of 1/2 multiplied by X in quantity, we see that this function is only going to be defined when the quantity underneath the radical is greater than or equal to 0, and that is true only when X is greater than or equal to 0. So when we're looking for points to plot for a function of F1, we're going to choose X values that are greater than or equal to 0. So we're going to start by looking at X equal to 0. So we're going to calculate F1 of 0. And this is going to be equal to the square root of the quantity of 1/2 multiplied by 0 in quantity, and this is going to be equal to 0 when we evaluate that expression. The next point that we'll want to look at is when X is equal to 2. So we'll calculate F1 of 2. That's going to be equal to the square root of the quantity of 1/2 multiplied by 2 in quantity. evaluating this expression, this is going to be equal to 1. Then finally we're gonna look at X equal to 4, so we'll calculate F1 of 4. And this is going to be equal to the square root of the quantity of 1/2 multiplied by 4 in quantity. And evaluating this expression, this is going to be equal to the square root of 2, which is approximately equal to 1.41. So now we want to do is plot these points on our graph. So we'll start off by looking at X equal to 0, which has a Y value of 0. The next point is going to be when X is equal to 2, our Y value is going to be equal to 1. And when we look at X equal to 4, our Y value is going to be equal to 1.41. So now what we'll do is connect these three points with a smooth curve. And that's going to give us our graph of F1. So now we need to do something similar for F2. If we look at a function F2 of X, it is going to be equal to the square root of the absolute value of the quantity of 1/2 multiplied by X in quantity. Since we have the absolute value underneath the radical, that means that the quantity underneath the radical is always going to be positive. So that means F2 is actually defined for all values of X. So we need to choose values of X that are both positive and negative. So we're gonna start off by choosing the same values for X that we initially did for F1. So we're going to look at F2 of 0, and this is going to be equal to the square root of the absolute value of the quantity of 1/2 multiplied by 0 in quantity, and evaluating this expression, this is going to be equal to 0. If we look at F2 of 2, This is going to be equal to the square root of the absolute value of the quantity of 1/2 multiplied by 2 in quantity, and this is going to be equal to 1. And if we look at F2 of 4, this is going to be equal to the square root of the absolute value of 1/2 multiplied by 4 in quantity. And this is going to be equal to the square root of 2, which is approximately equal to 1.41. So now we need to also choose negative values for X. So we'll choose X equal to -2, as our next point. So we'll calculate F2 of -2. And this is going to be equal to the square root of the absolute value of the quantity of 1/2 multiplied by minus 2 in the quantity. And this is going to be equal to one when we evaluate that expression. And if we look at F2 of -4, so choosing X equal to -4 as our next point, this is going to be equal to the square root of the absolute value of 12 of the quantity of quantity of 1/2 multiplied by -4 in quantity, and that is going to be equal to the square root of 2 when we evaluate that expression, and that's going to be approximately equal to 1.41. So now we can plot these points on our graph. To create our plot of F2. So, looking at the first point, we have when X is equal to 0, Y is going to be equal to 0, when X is equal to 2, or Y is going to be equal to 1, and when X is equal to 4, our Y value is going to be 1.41. When X is equal to -2, our y value is going to be equal to 1. And when X is equal to -4, our Y value is gonna be 1.41. So now we'll do is connect these points with a smooth curve. And in doing so, we'll see that for values of X greater than or equal to 0, that our two curves actually overlap. So, the functions F2 and F1 actually overlap for values of X greater than are equal to 0. So the graph that we have created on the screen is the graph that we were looking for for this problem. So now we want to turn our attention to describing how applying the absolute value function in F2 modifies the graph of F1. And for this, we see that the absolute value in F2 extends the domain of the function to include negative values of X, resulting in a graph of F1 that is symmetric about the y axis. So that is the second half of the answer to this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x)        ƒ₂(x)
          _
√ x         √ |x|

Key Concepts

Absolute Value Function

Graphing Functions

Transformation of Functions

Watch next

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.ƒ₁(x) ƒ₂(x) __ ___√ x √ |x|

Key Concepts

Absolute Value Function

Graphing Functions

Transformation of Functions

Watch next

Graph ƒ₁ and ƒ₂ together. Then describe how applying the absolute value function in ƒ₂ affects the graph of ƒ₁.

ƒ₁(x) ƒ₂(x)
_
√ x √ |x|