Algebra 2 Rational Functions

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what is a undefined or restricted value?

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1

what is a undefined or restricted value?

a value that makes the denominator = 0

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2

How do we find restricted values?

Set the denominator equal to 0 and solve

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3

Find the restricted value:

3x/x+5

x = -5

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4

Steps to simplify rationals

  • Factor both the numerator & denominator

  • cancel like factors from the numerator & denominator

  • write simplified answer

<ul><li><p>Factor both the numerator &amp; denominator</p></li><li><p>cancel <strong>like factors</strong> from the numerator &amp; denominator</p></li><li><p><strong>write simplified answer</strong></p></li></ul>
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5

Simplify.

x² + 3x

x² + 5x

x + 3 / x + 5

<p>x + 3 / x + 5</p>
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6
<p>Simplify.</p>

Simplify.

x(2x-3)

<p>x(2x-3)</p>
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7
<p>Simplify.</p>

Simplify.

2 / x + 3

<p>2 / x + 3</p>
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8

Steps to multiply rationals

  • DO NOT MULTIPLY. I REPEAT, DO. NOT. MULTIPLY

  • Factor numerators & denominators

  • cancel like factors and write remaining expression

<ul><li><p><strong>DO NOT MULTIPLY. I REPEAT, DO. NOT. MULTIPLY</strong></p></li><li><p><strong>Factor numerators &amp; denominators</strong></p></li><li><p><strong>cancel like factors and write remaining expression</strong></p></li></ul>
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9

Steps to divide rationals

  • DO NOT DIVIDE.

  • multiply by the reciprocal of the second rational expression

  • Follow process for multiplying

<ul><li><p><strong>DO NOT DIVIDE.</strong></p></li><li><p>multiply by the reciprocal of the second rational expression</p></li><li><p>Follow process for multiplying</p></li></ul>
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10
<p><strong>Multiply.</strong></p>

Multiply.

(x + 2) (x - 1) / x(x + 1)

<p>(x + 2) (x - 1) / x(x + 1)</p>
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11
<p><strong>Divide.</strong></p>

Divide.

(x - 8) ( x - 5) / (x + 7)²

<p>(x - 8) ( x - 5) / (x + 7)²</p>
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12

Steps to add rationals

  • ALWAYS make sure the denominators are the same

  • Then, add the numerators by combining like terms.

<ul><li><p><strong>ALWAYS</strong> make sure the denominators are the same</p></li><li><p>Then, add the numerators by <strong>combining like terms.</strong></p></li></ul>
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13

Steps to subtract rationals

  • Make sure the denominators are the same

  • Combine like terms, and don’t forget to distribute the negative sign if necessary!

<ul><li><p>Make sure  the denominators are the same</p></li><li><p>Combine like terms, and don’t forget to distribute the negative sign if necessary!</p></li></ul>
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14

To find common denominators…

  • Only use each factor once

  • factor each denominator individually & find the LCD

  • multiply them together to get a combined denominator

  • multiply the numerator by the respective denominator (see example)

<ul><li><p>Only use each factor once</p></li><li><p>factor each denominator individually &amp; find the <strong>LCD</strong></p></li><li><p>multiply them together to get  a combined denominator </p></li><li><p>multiply the numerator by the respective denominator (see example)</p></li></ul>
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15
<p><strong>Add. </strong></p>

Add.

3x + 2 / (x + 2)(x - 2)

<p>3x + 2 / (x + 2)(x - 2)</p>
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16
<p><strong>Subtract.</strong></p>

Subtract.

-4 / x + 2

<p>-4 / x + 2</p>
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17

Steps to solving rational equations

  • find common denominators (multiply rationals by a factor to make them all have the same denominator)

  • cross out the denominators and solve the remaining equation

  • check for extraneous (restricted values) solutions

<ul><li><p>find common denominators (multiply rationals by a factor to make them all have the same denominator)</p></li><li><p>cross out the denominators and solve the remaining equation</p></li><li><p>check for extraneous (restricted values) solutions</p></li></ul>
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18
<p><strong>Solve.</strong></p>

Solve.

x = -1

4 is extraneous.

<p>x = -1 </p><p>4 is extraneous.</p>
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19

vertical asymptote

set denominator = 0 and solve.

  • written as x = #

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20

domain

All real numbers except the vertical asymptote you found.

<p>All real numbers <strong>except the vertical asymptote you found.</strong></p>
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21

horizontal asymptote

Use the degrees of the numerator & denominator

  • equal degrees : coefficients of leading terms

  • numerator < < denominator : HA = 0

  • numerator > > denominator : NO HA

Written as y = #

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22

range

All real numbers except the horizontal asymtote you found. If no HA, then its all reals.

<p>All real numbers  <strong>except the horizontal asymtote you found</strong>. If no HA, then its all reals. </p>
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23

x intercept

set numerator = 0 and solve

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24

y intercept

plug in zero for all x-values

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25

holes

if you can cancel a factor from the numerator & denominator, then there is a hole at that x value.

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26

Find the characteristics:

x - 2 / x + 2

  • factored

  • VA: x = -2

  • Domain: R: x cant be -2

  • HA: y = 1

  • Range: R: x cant be 1

  • X-INT: (2,0)

  • Y-INT: (0,-1)

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