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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.$g(x)=|2 x-3|$

(a) $-\infty<x<\infty$(b)(c) $y \geq 0$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

Campbell University

Baylor University

University of Michigan - Ann Arbor

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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(a) determine the domain, …

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Find the domain and sketch…

for this problem. We've been given a function G of X equals the absolute value of two X minus three, and we have three goals for this problem. We want to find the domain and the range of G. And then we went to graph G because sometimes graphing the function gives us a visual confirmation that we have the correct domain and range selected. So let's start with domain domain. Are your inputs those of your ex values that you can put into a function and often for domain? Instead of trying to figure out which exes work? It's sometimes easier to figure out which exes don't work. Everything else will be in your domain. Well, this is an absolute value equation. I'm taking a number of times two minus three. Taking the absolute value. There are no exes that are excluded from this. You could make a positive negative zero a fraction. It doesn't matter. I can put it into this function and get out a perfectly valid answer. So my domain are all real numbers. X is anywhere from negative infinity to positive infinity, which I could write an interval notation that way. Okay, so domain is all real numbers. What about my range? My range are my Y values the numbers I get out from, ah, function Well, there is a limit here because absolute values on Lee give positive numbers. You know, at the smallest I could do zero if two X minus three equals zero, that gives me a zero output. But everything else is going to be bigger. So why has to be greater than or equal to zero? I can never get a negative number out of an absolute value function. So in interval notation, that means my range goes from zero with a square bracket to infinity. So let's graph this now. One thing I just said was, If what's inside those absolute values equals zero, then my value is zero. So let's see where that ISS if I let two X minus three equals zero. That gives me X equal. Three halves so at X equal in three halves, my output is going to be zero. So let's pick a few values of X, plug them in and see what we get. Let's let X equal three. That's gonna be six, minus three or three. How about X equaling five Well, 10 minus three. That's 71234567 456 Counted that right. Okay, What about some numbers smaller than my point? What if x zero well, absolute value of negative three would be three. And what effects is negative too? Well, that gives me negative four minus three. That's negative. Seven absolute value is positive. Seven. So, as you can see, I've got a symmetric V shape to my graph with a point right there at X at X, equaling three haps. I have nothing down here. So I do have the correct range. Why has to be zero or bigger? And my domain is all re Alexis. So my domain and range and sketch for my given function.

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