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In Exercises 1–3, the graph of a quadratic function ƒ is given. Use the graph to find the solution set of each equation or inequality. See Example 1.(a) $x^{2}-4 x+3=0$(b) $x^{2}-4 x+3>0$(c) $x^{2}-4 x+3<0$

$\begin{array}{ll}\text { (a) }\{1,3\} & \text { (b) }(-\infty, 1) \cup(3, \infty) \text { (s) }(1,3)\end{array}$

Precalculus

Algebra

Chapter 11

Quadratic Equations, Inequalities, and Functions

Section 8

Polynomial and Rational Inequalities

Introduction to Conic Sections

Equations and Inequalities

Functions

Polynomials

Missouri State University

Harvey Mudd College

Baylor 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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All right, So we have the function X squared minus four X plus three. And we want the solutions to this. Well, given that we have this graph here, it seems pretty obvious then that we have solutions, right? We are intersecting the X axis when X equals one when X equals three. Okay, someone's not so bad. We just basically have to look at the graph and pick out the roots. So those air the solutions to the equation now, similarly, I will go ahead and redraw this. We're gonna take the same quadratic, but we want now to look at when x squared minus four X plus three is greater than zero. We have the same routes, recurring at one and at three. But now we want to know when X is greater than zero. Okay. Or when y is greater than zero, I should say, right when the function is greater than zero. Well, if we're at zero on the access, then we're a greater than zero. When were above the access rights on this portion of the graph. Right here. We're above the access. So where is that happening? Well, that's happening. When were to the left of one and one more to the right of three. So we would say that if this function is greater than zero, that X it's gonna be less than one or X is greater than three or using interval notation, which is probably should be using. This would be the interval from negative infinity to one or three to infinity. And if we take the same curve and we want to examine when X squared minus four X plus three is less than zero. Well, we can have the same discussion. We have a route one in a rude at three. When are we less than zero? Well, we're gonna be less than zero when we are below the axis. So in other words, in this stretch right here, where are we below the axis? 10. Well, we're below the axis on this interval between the one and the three. So we would say our solution set then is the interval from 1 to 3. So that's our solution sets for when the functions equals zero when it's greater than zero and when it's less than zero

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