Find the domain and range and sketch the graph of the function h(x)=√(4-x^2)

Name: Problem 30, Chapter 1: Essential Calculus Early Transcendentals
Uploaded: 2020-05-17T02:33:59-08:00
Duration: 10 min 40 s
Channel: Sushmit Acharya
Description: Find the domain and range and sketch the graph of the function h(x)=√(4-x^2)

Find the domain and range and sketch the graph of the...

Key Concepts

Interplay Between Algebraic Expressions and Geometric Shapes

Many functions, particularly those defined by square root expressions, have graphs that can be related to common geometric figures. Recognizing these shapes helps in visualizing and sketching the function's graph, as the algebraic conditions often correspond to defining parts of geometric figures.

Graphing Radical Functions

Graphing functions that involve radicals typically involves understanding key transformations and restrictions imposed by the radical. The process includes identifying symmetry, intercepts, and boundary points derived from the domain restrictions. This understanding aids in producing an accurate visual representation of the function.

Domain of a Function

This concept involves identifying all the possible input values (x-values) for which a function is defined. In problems involving square root functions or any operation with inherent restrictions (like division by zero), the domain is determined by the conditions that ensure the function's expression is mathematically valid.

Range of a Function

The range of a function is concerned with all possible output values (y-values) that the function can produce based on its definition and the allowed domain. For functions involving square roots, since the square root operation yields only nonnegative outputs, the method of determining the range focuses on how the expression inside the square root affects the possible outcomes.

Non-Negativity Requirement for Radicands

When dealing with square root functions, the expression under the square root, known as the radicand, must be nonnegative to produce real number outputs. This requirement is key to identifying the domain of such functions and plays a significant role in determining the resulting graph and range.

Transcript

00:01 So here we start off with a function h of x equals square root of 4 minus x squared and we're finding the domain range and graphing the function as well.

00:09 So since this is a square root function, we have to make sure that the function inside the square root is greater than zero because obviously we can't take a square root of a negative number.

00:25 So that's a general rule of thumb for finding domain of domain or range of a function that's inside a square root.

00:34 So we basically say 4 minus x squared is greater than or equal to 0.

00:39 We subtract 4 on each side.

00:43 Then we get negative x squared is greater than or equal to negative 4.

00:47 We try to get rid of the negatives.

00:50 We have a negative 1.

00:54 Because we're dividing by negative 1, it is essential that we, switch the signs.

01:01 After that, it's going to become x squared is a less than or equal to the 4.

01:06 This is very important.

01:09 Very important.

01:10 Let me start that.

01:12 All right.

01:13 Then, since it's a x squared is less than or equal to 4, we can't just take the square root and then just get 2 like we've done here.

01:25 Because if you think about it, negative 2.

01:30 2 squared also equals 4 and 2 squared also equals 4.

01:35 So a little tip for whenever you have a variable that is raised to an even exponent, you can do negative times the variable.

01:48 Quantity squared like right here is less than or equal to 4, and then x squared is less than or equal to 4.

01:57 Then after that, we take the square root of both sides.

02:01 In this case, we get negative x is less than or equal to 2.

02:09 Again, we divide it by negative 1, and then we get x is greater than or equal to negative 2.

02:16 As i said before, remember to change the sign when dividing or multiplying by a negative value.

02:23 On this side, we have x squared is less than or equal to 4.

02:28 We take the square root of both sides.

02:31 Then we get x is less than or equal to two.

02:37 So from these two answers, we can deduce that negative 2 is less than or equal to x, which is less than or equal to 2, which is our domain.

02:51 So what we did here is get the function and then basically find what could x be.

03:01 So domain is always what could x be, which is the independent variable.

03:06 In this case.

03:09 So we're solving for x whenever we're trying to find what the domain is.

03:17 And from that, we say, from that, remember the fact that anything inside the radical must be greater than or equal to zero.

03:29 Then we solve for x as we just did.

03:32 And then we eventually got to the answer, which was negative 2, comma 2, or x has to.

03:39 To be between negative 2 and 2.

03:43 All right, so now we're trying to find the range.

03:47 One main thing about the range, since domain is the values x can be, range is the value, are the values that y can be, or in this case, h of x can be right here.

04:05 Here, i said that h of x is greater than or equal to 0.

04:10 Why? since the function is in a square root, we know that a square root of any number, whatever it is, cannot be less than zero.

04:22 And we make sure that we know that fact and we move forward with it.

04:28 Also, i just replaced h of x with y.

04:32 You can do that depending when you're solving problems.

04:35 It means the same thing.

04:37 So i restated the function again.

04:40 And then i said, h of x equals square root of.

04:44 4 minus x squared.

04:46 I changed the h of x into y and i squared both sides in order to get rid of the square root and then i said y squared minus 4 equals negative x squared so i basically subtracted 4 on each side and then from there we went to dividing each side by negative 1 and which led us to x squared equals 4 minus y squared.

05:18 Then to get rid of the squares, we just take the square root of both sides.

05:23 Then we get x equals square to 4 minus y squared.

05:27 So here we're solving for x.

05:33 Then after solving for x, we say, as we did before while trying to find the domain, here we say 4 minus y squared is greater than or equal to 0.

05:45 As before, because this function, is inside the square root, we know that it has to be greater than or equal to zero.

05:54 Then we subtract four from each side...

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