Difference between revisions of "004 Sample Final A, Problem 1"

Revision as of 19:23, 28 April 2015

Find $f^{-1}(x)$ for $f(x)={\frac {3x-1}{4x+2}}$

Foundations
How would you find the inverse for a simpler function like $f(x)=2x+4$
Answer:
You would replace $f(x)$ with $y$ . Then, switch $x$ and $y$ . Finally, we would solve for $y$ .

Solution:

Step 1:
We start by replacing $f(x)$ with <y>.
This leaves us with $y={\frac {3x-1}{4x+2}}$

Step 2:
Now, we swap $x$ and $y$ to get $x={\frac {3y-1}{4y+2}}$ .

Step 3:
Starting with $x={\frac {3y-1}{4y+2}}$ , we multiply both sides by $4y+2$ to get
$x(4y+2)=3y-1$ .
Now, we need to get all the $y$ terms on one side. So, adding 1 and $-4xy$ to both sides we get
$2x+1=3y-4xy$ .

Step 4:
Factoring out $y$ , we get $2x+1=y(3-4x)$ . Now, dividing by $(3-4x)$ , we get
${\frac {2x+1}{3-4x}}=y$ . Replacing $y$ with $f^{-1}(x)$ , we arrive at the final answer
$f^{-1}(x)={\frac {2x+1}{3-4x}}$

Final Answer:
$f^{-1}(x)={\frac {2x+1}{3-4x}}$

@@ Line 4: / Line 4: @@
 ! Foundations
 |-
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+| How would you find the inverse for a simpler function like <math>f(x)=2x+4</math>
 |-
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+|Answer:
 |-
-|Answers:
+|You would replace <math>f(x)</math> with <math>y</math>. Then, switch <math>x</math> and <math>y</math>. Finally, we would solve for <math>y</math>.
-|-
-|
-|-
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 |}
@@ Line 21: / Line 17: @@
 ! Step 1:
 |-
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+|We start by replacing <math>f(x)</math> with <y>.
 |-
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+|This leaves us with <math>y=\frac{3x-1}{4x+2}</math>
 |}
@@ Line 29: / Line 25: @@
 ! Step 2:
 |-
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+|Now, we swap <math>x</math> and <math>y</math> to get <math>x=\frac{3y-1}{4y+2} </math>.
-|-
-|
 |}
@@ Line 37: / Line 31: @@
 ! Step 3:
 |-
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+|Starting with <math>x=\frac{3y-1}{4y+2} </math>, we multiply both sides by <math>4y+2</math> to get
 |-
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+|<math>x(4y+2)=3y-1</math>.
 |-
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+|Now, we need to get all the <math>y</math> terms on one side. So, adding 1 and <math>-4xy</math> to both sides we get
+|-
+|<math> 2x+1=3y-4xy</math>.
 |}
@@ Line 47: / Line 43: @@
 ! Step 4:
 |-
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+|Factoring out <math>y</math>, we get <math> 2x+1=y(3-4x) </math>. Now, dividing by <math>(3-4x)</math>, we get
+|-
+|<math>\frac{2x+1}{3-4x}=y</math>. Replacing <math>y</math> with <math>f^{-1}(x)</math>, we arrive at the final answer
 |-
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+|<math>f^{-1}(x)=\frac{2x+1}{3-4x}</math>
 |}
@@ Line 55: / Line 53: @@
 ! Final Answer:
 |-
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+|<math>f^{-1}(x)=\frac{2x+1}{3-4x}</math>
 |}
 [[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]