Difference between revisions of "005 Sample Final A, Question 4"

Latest revision as of 20:16, 21 May 2015

Question Find the inverse of the following function $f(x)={\frac {3x}{2x-1}}$

Foundations:
1) How would you find the inverse for a simpler function like $f(x)=3x+5$ ?
Answer:
1) you would replace f(x) by y, switch x and y, and finally solve for y.

Step 1:
Switch f(x) for y, to get $y={\frac {3x}{2x-1}}$ , then switch y and x to get $x={\frac {3y}{2y-1}}$

Step 2:
Now we have to solve for y: ${\begin{array}{rcl}x&=&{\frac {3y}{2y-1}}\\x(2y-1)&=&3y\\2xy-x&=&3y\\2xy-3y&=&x\\y(2x-3)&=&x\\y&=&{\frac {x}{2x-3}}\end{array}}$

Final Answer:
$y={\frac {x}{2x-3}}$

@@ Line 2: / Line 2: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! Foundations:
 |-
-|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
+|1) How would you find the inverse for a simpler function like <math>f(x) = 3x + 5</math>?
 |-
-|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
+|Answer:
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
+|1) you would replace f(x) by y, switch x and y, and finally solve for y.
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Step 1:
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+| Switch f(x) for y, to get <math>y = \frac{3x}{2x-1}</math>, then switch y and x to get <math>x = \frac{3y}{2y-1}</math>
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Step 2:
 |-
-|e) True.
+| Now we have to solve for y:
+::<math> \begin{array}{rcl}
+x & = & \frac{3y}{2y-1}\\
+x(2y - 1) & = & 3y\\
+xy - x & = & 3y\\
+xy - 3y & = & x\\
+y(2x - 3) & = & x\\
+y & = & \frac{x}{2x - 3}
+\end{array}</math>
+|}
+{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+! Final Answer:
 |-
-|f) False.
+|<math> y = \frac{x}{2x-3}</math>
 |}