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(a) determine the domain, (b) sketch the graph and (c) determine the range of the function defined by the given equation.$w(x)=\sqrt{3 x-4}$

(a) $x \geq 4 / 3$(b)(c) $y \geq 0$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Missouri State University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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for this problem, we have been given a function w of X equal in the square root of three X minus four. Now, for this problem, we're gonna find three things. We're going to find the domain and the range for the function, and they were gonna graph the function and graphing a function is often a good visual representation of both domain and range. And make sure that we have everything correct. So first, let's try to figure out our domain Read domain is your input those your ex values that you can put into your function? So a good way to figure out domain is to figure out which exes are excluded, and then everything that remains must be in your domain. Well, my function is a square root, so there are some numbers that are excluded when I have a square root. What's underneath that radical side has to be greater than or equal to zero. If it's a negative, it means that we're imaginary numbers and we're not using those right now. So let's solve for X three X is greater than or equal to four X is greater than or equal to four thirds. So there's my domain. If I want to write this in interval notation, I can write this as four thirds to infinity. Okay? So as long as X is bigger than or equal to four thirds, I'm gonna have a positive number under my radical on. I don't care what the positive number is. I could always take the square root. What about range? Well, my functions a square root, all square roots return positive numbers. So my range why is gonna have to be greater than or equal to zero square roots? Those radicals cannot return a negative number. So how does that look when we graph this? Well, x equally four thirds. That's about here. And that was our, um, smallest X we could have. And at that point, that gives us a zero under the radical value of zero For why? Well, let's just kind of pick out some numbers and see what we have. What if I let x equal? Um, has to be bigger than four thirds. Let's let X equal three nine minus four. Well, let's see. 1239 minus four is five squared of five is between two and three. I could put a dot there. Let's let x equal. Um, sure, it's not equal. 10. That's 30. 30 minus four is 26. So that's about five. So 12345612345 That's gonna go there. And you can just see from that this is gonna look like a standard square roots thing. It has a starting point and a gentle slope upward so you can see the X and Y will both be increasing without bounds. Ex beginning at four thirds. Why, beginning at zero. So sketch domain and range for are given function.

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