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Consider the function defined by the equation $f(x)=\frac{1}{2 x^{2}-x-3}$ (a) Determine its domain, and (b) Sketch its graph clearly showing all asymptotes. (c) Find its range by solving for $x$ as a function of $y$. (d) Using (c), locate the coordinates of the "turning point" of the function.

(a) $x \neq-1,3 / 2$(c) $y \leq-8 / 25$ or $y>0$(d) $(1 / 4,-8 / 25)$

Algebra

Chapter 1

Functions and their Applications

Section 7

More on Functions

Functions

Missouri State University

Harvey Mudd College

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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we can factor the denominator into two parts and because the denominator cannot be zero. So the mind of this function is X does not equal to minus one and ex. That's not legal to three halfs. To find the vertical Essman thought of this function, we need the denominator to be their role and the numerator does not. So X equals minus one and X equals three half. Satisfy this condition to find the horizontal Essman thought we need to find the limit of this function as X goes to infinity. And it's quite easy for us to see that this limit is terrible and the horizontal s mom taught is y equals zero for Per Say we need to find a range of this function. But first notes that the denominator of these functions a quadratic form and we can rewrite this quadratic form like this. Since the square of some number cannot be negative, this quadratic form must be greater than or equal to minors 25/8. And this function is just as a reciprocal of this quadratic form. If the quadratic form takes value between zero and positive infinity, it's reciprocal also takes value between zero and positive infinity. And if this radical form takes value between minus 25/18 0, it's really it's reciprocal. Will take value between minus infinity and minus 8/25. And hence this is just a range of this function for party. We need to find the turning point, not there. For this quadratic form. It has a minimal at X equals 1/4 and this local minimal is minus 25/8 and avoids reciprocal. It has no go minimal at the same point. And this point is 1/4 and it's reciprocal as minus eight or 25 hence this is a turning point of the function.

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