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### Course: Algebra (all content)>Unit 7

Lesson 29: Comparing features of functions (Algebra 2 level)

# Comparing functions: shared features

Sal is given the formula of one function and the graph of the other, and he determines which features are shared by both functions. Created by Sal Khan.

## Want to join the conversation?

• what is g(-4)
(1 vote)
• If g(x) = x+2 (I can presume that by looking at the graph of the function g in the video, supposedly the pink line is g(x)'s graph but oddly enough the line passes through -3 so it's most likely not equal to x+2) then g(-4) = -4 + 2 = -2
Jacob
• About "end behavior the same" :
I guess this is a minor point, but although both functions get infinite as |x| does, since f is more or less quadratic (cubic minus linear) it moves quite a bit faster w.r.t. |x| than g. Is there some mathematical reason for thinking of them as equivalent?
• so i think this is the first time in the playlist i've heard that relative maximum and local maximum are the same thing. is that always true for all situations? since it wasn't the goal of this video to explain that, i just want to be sure. :D thx!
(1 vote)
• As far as I understood, yes, these are the same. Sal clearly deals with relative maximum as he would deal with a local one. Actually, I have never heard local maximums to be called relative, so it even sounded a bit weird. However, I am Ukrainian, and here we actually don`t use a term "local maximum (if translated)" - they are "relative" ones in Ukrainian. I`m just trying to prove my argument from different points of view, even using etymology of the words. Hope this helps.
• Are periodic functions the same as odd function?
Thanks.
• No. The parity is a different property from periodicity. Some periodic functions are odd, some are even, some are neither.
sin, tan, csc and cot are odd
cos and sec are even

As for other periodic functions, some are odd, some are even, some are neither. For example e^(i*x) is periodic, but it is neither odd nor even.

Also note that when you add or subtract to a trigonometric function, depending on what you add, you may alter its parity. While sin x is odd, sin (x+1) is neither and sin(x + 7π/2) is even.

Finally, odd or even functions do not have to be periodic. y=x² is even, y=x³ is odd, but neither of these is periodic.
• The practice question has me totally confused. I understand the concept but the description of f(x) makes no sensef is a function defined on all integers. A verbal definition of f is given below.
g is a function defined on all real numbers. Its graph in the range −10≤x≤10 is given below. Suppose the trend suggested by the graph's edges continues.
Which of the features are shared by f(x) and g(x)?

Select all that apply.
•They have the same y-intercept.
•They are both even.
•They have the same minimum value.
•They are both periodic.

Definition for f
• If the remainder of x divided by 3 is 0, then f(x)=11.
• If the remainder of x divided by 3 is 1, then f(x)=−7.
• If the remainder of x divided by 3 is 2, then f(x)=2.

What the heck is the remainder of x divided by 3? What is this. I can't find anything worded similarly. Is this using the remainder theorem?
• how do i know the end behavior ? what kind of end behaviors could be there ? is there any formula ?
• I'm confused on how to find a x-intercept of a function.
Is this how you do it?:
f(x)=x+6-9
x=3
f(x)=3+6-9
(Because in order to get f(x) to =0,x is 3)
(1 vote)
• That's correct. f(x) intercepts the x axis when y = 0. So we set f(x) = 0 which is the same as x+6-9 = 0. Now solve for x, which leaves us with x = 3.
• What is a periodic function?
Where can I find the video about it?
(1 vote)
• Search up periodic function in the KA search bar.
A periodic function is a function that repeats its outputs every couple inputs. Therefore, different intervals of the domain graph identical outputs. A simple example is the sine wave. The wave repeats in the exact same way as you go sideways. In function notation, a function is periodic if and only if:
f(x + P) = f(x)
For some constant P. P, in the case a wave function, would be the wavelength of its graph. The sine wave has a wavelength of 2π. Therefore we can write:
sin(x + 2π) = sin(x)
• For the following question from the practice, would somebody please explain f(x) to me? I can see the highest point is 11, lowest point is -7, but how can I answer anything else with the information they have provided?

f is a function defined on all integers. A verbal definition of f is given below.
g is a function defined on all real numbers. Its graph in the range −10≤x≤10 is given below. Suppose the trend suggested by the graph's edges continues.
Which of the features are shared by f(x) and g(x)?

Select all that apply.
•They have the same y-intercept.
•They are both even.
•They have the same minimum value.
•They are both periodic.

Definition for f
• If the remainder of x divided by 3 is 0, then f(x)=11.
• If the remainder of x divided by 3 is 1, then f(x)=−7.
• If the remainder of x divided by 3 is 2, then f(x)=2.
(1 vote)
• Because you don't have the graph, I cannot answer the full question, but I can describe f(x). Let's think. *If the remainder of x/3 is 0, then f(x)=11.* If the remainder is 0, then x is divisible by 3. So, for x=a multiple of 3, f(x)=11. *If the remainder of x/3 is 1, then f(x)= -7.* This means that, if x is one above a multiple of 3, then f(x)= -7. *If the remainder of x divided by 3 is 2, then f(x)-2.* This means that, if x is two above a multiple of 3, then f(x)=2. The graph of f(x) would then be just a series of points at where x is an integer, with no line connecting them.
(1 vote)
• where can you learn about cubic functions?
(1 vote)

## Video transcript

Which of the features are shared by f of x and g of x? Select all that apply. So they give us f of x as being defined as x to the third minus x. And they define g of x, essentially, with this graph. So what are our options? So the first one is that they are both odd. So just by looking at g of x, you can tell that it is not odd. The biggest giveaway is that an odd function would go through the origin. G of zero would have to be equal to zero. If you want to go straight to the definition of an odd function, g of x would have to be equal to the negative of g of negative x. So for example, g of 3 looks like it is 4. g of 3 is equal to 4. In order for it to be odd, g of negative 3 would have to be equal to negative 4. But we see that g of negative 3 is not equal to negative 4. So this one is definitely not odd. So this statement can't be true. They both can't be odd. So that's not right. They share an x-intercept. So g of x only has one x-intercept. It intersects the x-axis right over here at x equals negative 3. Now let's think about the x-intercepts of f of x. And to do that, we just need to factor this expression, f of x is equal to x to the third minus x, which is the same thing if we factor an x out of x times x squared minus 1. X squared minus 1 is the difference of squares. So we could rewrite that as-- so we'll write our x first, this x. And then x squared minus 1 is x plus 1 times x minus 1. So when does f of x equal 0? Well, f of x is equal to 0 when x is equal to 0. When x is equal to 0, that would make this entire expression 0. When x is equal to negative 1, that would make this term, and thus the entire expression, 0. And when x is equal to positive 1, that would make this last part zero, which would make this entire product 0. So here are the zeroes of f of x, and none of these coincide with the zeroes of g of x. So they don't share an x-intercept. They have the same end behavior. Now this is interesting. This is saying what's happening as x gets really, really, really, really large, or as x gets really, really, really, really, really small. So we could just think about it right over here. As x gets really, really, really, really, really large, this x to the third is going to grow much faster than this x term right over here. So as x grows really, really, really, really large, f of x is going to grow really, really, really large. So the graph-- and I don't know exactly, to see if I could plot a couple of other points-- but the bottom line is f of x is going to approach infinity as x approaches infinity, or f of x approaches infinity as x approaches infinity, or as x gets larger and larger and larger. And then what happens as x gets smaller and smaller and smaller? If we have really small values of x-- so really negative values of x, I should say-- once again, this, right over here, is going to dominate. So f of x is going to become really negative. So f of x is going to approach negative infinity, as x approaches negative infinity. And that is the same behavior of g of x. As x approaches a really large value, g of x approaches a really large value, maybe not as fast as f of x, but it still approaches it. And likewise, as x decreases, so does g of x decrease. It doesn't decrease as fast as f of x might, but it's still going to decrease. So they do seem to have the same end behavior, at least based on the way that we thought about it just now. Now the last option is they have a relative maximum at the same x value. So we have to think about what the maximum points are. Well, actually, we already know that this is not true, because g of x has no relative maximum points. In order to have a maximum point, you would have to do something like this. This, right over here, would be a relative maximum, or, you could say, a local maximum point. It's larger than all of the points around it, but eventually the function does surpass it. But this, right over here, has no local maxima, or relative maxima, or little bumps in it. g of x doesn't have any. So they can't have relative maximum at the same value. So this, also, is not an option.