Which one of the followings are functions? If so then find its Domains and Ranges. If not, then

Which one of the followings are functions? If so then find...

Key Concepts

Function

A function is a relation between a set of inputs and a set of outputs where each input is associated with exactly one output. This key concept underlies all analysis when determining whether a given equation defines a function.

Domain

The domain of a function is the set of all possible input values for which the function is defined. This concept is essential because it outlines the permissible values that can be substituted into the equation.

Range

The range of a function is the set of all output values produced by the function as the input varies over its domain. Understanding the range is crucial for capturing the behavior of the function with respect to its outputs.

Vertical Line Test

The vertical line test is a graphical method used to determine if a relation is a function. If any vertical line intersects a graph at more than one point, then the relation fails to be a function because it assigns more than one output to a single input.

Linear and Constant Functions

Linear functions, such as those expressed in the form y = mx + c, and constant functions like y = 0 or y = 5, are foundational examples in algebra. They typically have well-defined, continuous domains and simple ranges, illustrating basic principles of functional relationships.

Graphical Representation of Equations

Many algebraic equations represent curves on a coordinate plane. Analyzing these graphs helps in determining whether an equation defines a function and aids in identifying the domain and range. This concept applies to various types of curves, including lines, parabolas, circles, ellipses, and hyperbolas.

Conic Sections

Conic sections, which include circles, ellipses, parabolas, and hyperbolas, are curves formed by the intersection of a plane with a cone. Understanding their properties and symmetry is essential when assessing if they can represent functions over their entire domain or if restrictions are needed to meet the function criteria.

Transcript

00:01 So in this question, we're going to have four different pairs of functions for which to consider the domains and ranges.

00:08 So in part a, we're considering the graph of y equals x squared minus nine to begin.

00:17 Well, what should this look like? if i think about this guy graphically, this is just your standard parabola, y equals x squared, shifted down by nine units, right? so down here would be 0 comma negative 9.

00:32 You would happen to have x intercepts at 3 and negative 3.

00:39 And you would have a graph that looks roughly like this.

00:42 Yes, here is y equals x squared minus 9.

00:47 What would i say about the domain of this? well, the domain of any polynomial function is all reals, negative infinity to positive infinity.

00:56 How about my range? well, my range starts with a smallest y value of negative 9 and goes up from there forever and ever.

01:07 So my range would be from negative 9 to positive infinity.

01:11 Now, let's compare that with y equals the square root of x squared minus 9 and get its domain and range, a little more interesting.

01:24 So how would i get the domain? well, in order to be in the domain of this function, i need x squared minus 9 to be greater than or equal to 0, right? i can only take the square root of something non -negative.

01:37 What does that imply? well, that implies that x squared is greater than or equal to 9, correct? now, i can take the square root of each side, but i've got to be extremely careful when i do.

01:52 Remember, the square root of x squared is not just x, but instead it's the absolute value of x.

02:03 And so what i'm getting is my domain is when the absolute value of x is greater than or equal to 3.

02:11 I'm getting x is greater than or equal to 3 or x is less than or equal to negative 3.

02:21 My domain would be in interval notation, negative infinity to negative three, as well as from three to positive infinity.

02:34 Now, what about the range of this guy? well, my range, remember, i can only take the square root of something that is greater than or equal to zero.

02:46 That was for my domain, but the output of a square root function.

02:50 What comes out must always be, greater than or equal to zero.

02:55 So my range this time is from zero to infinity.

03:02 You can get zero as an output.

03:04 For example, if x was three, three squared minus nine, i'd have nine under the radical.

03:11 9 minus nine is zero.

03:13 Squared of zero is zero.

03:15 So my range is zero to infinity.

03:18 Next, we'll head on to part b.

03:23 In part b, we'll start by considering y equals 2 minus x squared.

03:30 Well, what's this going to look like? this would be a quadratic opening down due to the negative leading coefficient, the negative coefficient in front of the x squared.

03:41 It's opening down, and it's been shifted up two units, right? some people might like to rewrite this as negative x squared plus two.

03:51 I have a y intercept at 2.

03:56 I'm opening down.

03:58 I have x intercepts of plus or minus root 2, and that's between 1 and 2 and negative 1 and negative 2.

04:08 Here's the graph of y equals 2 minus x squared.

04:13 And so if i want the domain of this guy, the domain of this quadratic is all reels.

04:20 And my range, the set of ys that can come out of this, well, they are from negative infinity to positive 2, inclusive of the positive 2.

04:31 But now, let's move on to our radical function, our y equals the square root of 2 minus x squared.

04:41 What's going to have to be true for me to be in the domain of this function? i can only take the square root of something that is greater than or equal to zero.

04:51 So i need 2 minus x squared to make greater than or equal to zero.

04:58 This implies that negative x squared is greater than a equal to zero.

05:02 To negative 2.

05:06 I'll divide both sides by negative 1.

05:10 Getting x squared is less than or equal to 2.

05:15 I can extract square roots.

05:18 Now, when i do, again, i have to be careful.

05:22 The square root of x squared is not just x, but it's the absolute value of x.

05:27 My absolute value of x is less than or equal to root 2.

05:33 That's equivalent to saying that x itself is between negative 2 and positive 2, negative square root of 2 and positive square of 2, excuse me, inclusive.

05:49 And so my domain here is in fact, from negative square root of 2 to positive square to 2.

05:56 And that will be my final answer for that domain.

06:00 How about my range? well, again, when i take the square root of some quantity, the result is always greater than or equal to zero...

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