# Algebra 2 problems, need help

• Aug 14th 2008, 04:25 PM
Kingreaper
Algebra 2 problems, need help
Ok, these are problems from a summer packet I have to turn in for precalc honors. The problems are based around algebra 2.
Find the inverses, f^-1(x) of the following functions using 3 different methods.

f(x)=x^2 +2x -5

f(x)=log_9e^sin(x/2) <---I really need help with this one.

f(x)=2x+1
3-x

Solve each of the following systems algebraically and using matrices
√3x - 2y = -1

2x + 5y^2 = 12
• Aug 14th 2008, 08:24 PM
Soroban
Hello, Kingreaper!

Do you know how to find an inverse function?

Quote:

Find the inverses, $\displaystyle f^{\text{-}1}(x)$ of the following functions
. . using 3 different methods.
. . I don't know three different methods!

$\displaystyle f(x)\:=\:x^2 +2x -5$

(1) Replace $\displaystyle f(x)$ with $\displaystyle y\!:\;\;y \;=\;x^2+2x-5$

(2) Interchange $\displaystyle x$'s and $\displaystyle y$'s: . $\displaystyle x \;=\;y^2 + 2y - 5$

(3) Solve for $\displaystyle y$

. . We have: .$\displaystyle y^2 + 2x - (x+5) \;=\;0$

. . Quadratic Formula: . $\displaystyle y \;=\;\frac{\text{-}2\pm\sqrt{2^2 + 4(x+2)}}{2} \;=\;\frac{\text{-}2 \pm\sqrt{4x+12}}{2}$

. . . . $\displaystyle y \;=\;\frac{\text{-}2 \pm2\sqrt{x+3}}{2} \;=\;\text{-}1 \pm\sqrt{x+3}$

(4) Replace $\displaystyle y$ with $\displaystyle f^{\text{-}1}(x)\!:\;\;f^{-1}(x) \;=\;-1 \pm\sqrt{x+3}$

Note that this inverse is not truly a function.

Quote:

$\displaystyle f(x)\:=\:\log_9\!\left[e^{\sin\frac{x}{2}}\right]$

$\displaystyle (1)\;\;y \;=\;\log_9\!\left[e^{\sin\frac{x}{2}}\right]$

$\displaystyle (2)\;\;x \;=\;\log_9\!\left[e^{\sin\frac{y}{2}}\right]$

$\displaystyle (3)\;\;\log_9\!\left[e^{\sin\frac{y}{2}}\right] \;=\;x$

. . . . . . $\displaystyle e^{\sin\frac{y}{2}} \;=\;9^x$

. . . . . . $\displaystyle \sin\frac{y}{2} \;=\;\ln\left(9^x\right) \;=\;x\!\cdot\!\ln 9$

. . . . . . . . $\displaystyle \frac{y}{2} \;=\;\arcsin\left(x\!\cdot\!\ln 9\right)$

. . . . . . . . $\displaystyle y \;=\;2\!\cdot\!\arcsin\left(x\!\cdot\!\ln9\right)$

$\displaystyle (4)\;\;f^{-1}(x) \;=\;2\!\cdot\!\arcsin\left(x\!\cdot\!\ln9\right)$

Quote:

$\displaystyle f(x) \:=\:\frac{2x+1}{3-x}$

$\displaystyle (1)\;\;y \;=\;\frac{2x+1}{3-x}$

$\displaystyle (2)\;\;x \;=\;\frac{2y+1}{3-y}$

$\displaystyle (3)\;\;\frac{2y+1}{3-y} \:=\:x \quad\Rightarrow\quad 2y+1\:=\:x(3-y) \quad\Rightarrow\quad 2y + 1 \:=\:3x - xy$

. . $\displaystyle 2y + xy \:=\:3x - 1 \quad\Rightarrow\quad y(2+x) \:=\:3x-1 \quad\Rightarrow\quad y \:=\:\frac{3x-1}{x+2}$

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

Is there a typo in the last problem?
Matrices work only on linear systems of equations.

• Aug 16th 2008, 09:49 AM
Kingreaper
Thank you so much, but I have a few more questions. First, how do you type the equations so well, with the powers in the right place without the ^? Second, how do you do the non-linear problem algebraically instead of with matrices? Also, here is another I don't understand.
Find f times g, f/g, f of g, g of f

f(x)=6√x + x^2
5th root (x-2)

g(x) 3
5(3rd root(x^2))
• Aug 16th 2008, 08:00 PM
ibnashraf
Quote:

Originally Posted by Kingreaper

f(x)=6√x + x^2
5th root (x-2)

g(x) 3
5(3rd root(x^2))

hope the following are the equations u referring to above ......

$\displaystyle f(x) \:=\: \frac {6\sqrt x + x^2}{\sqrt[5]{x - 2}} \quad\Rightarrow\quad \frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}}$

$\displaystyle g(x) \:=\: \frac {3}{5\sqrt[3]{x^2}} \quad\Rightarrow\quad \frac {3}{5x^\frac{2}{3}}$

Quote:

Originally Posted by Kingreaper
how do you type the equations so well, with the powers in the right place without the ^?

first time i'm practicing typing out this equation stuff as well ..... there is a forum called 'laTex help' under 'maths help forum lounge'. u may want to check that out to understand what's going on.
• Aug 16th 2008, 09:05 PM
Kingreaper
yes, those are the equations, lol. Thanks for the latex tip, I'm gonna try that now.
• Aug 16th 2008, 09:05 PM
ibnashraf
Quote:

Also, here is another I don't understand.
Find f times g, f/g, f of g, g of f

ok i'll try and give this a shot to get some more practice with this laTex programming thing ....

$\displaystyle f(x) \times g(x) \:=\: \frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} \times \frac {3}{5x^\frac{2}{3}} \quad\Rightarrow\quad \frac {18x^\frac{1}{2} + 3x^2}{5x^\frac{2}{3}(x-2)^\frac{1}{5}}$ etc

and

$\displaystyle \frac{f(x)}{g(x)} \:=\: \frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} \times \frac {5x^\frac{2}{3}}{3} \quad\Rightarrow\quad \frac {30x^\frac{7}{6}+5x^\frac{8}{3}}{3(x-2)^\frac{1}{5}}$ etc

Also;

$\displaystyle fg(x) = f\left[\frac {3}{5x^\frac{2}{3}}\right] \quad\Rightarrow\quad \frac{6\left[\frac {3}{5x^\frac{2}{3}}\right]^\frac{1}{2}+\left[\frac {3}{5x^\frac{2}{3}}\right]^2}{\left[\frac {3}{5x^\frac{2}{3}} - 2\right]^\frac{1}{5}}$ etc

and

$\displaystyle gf(x) = g\left[\frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} \right] \quad\Rightarrow\quad \frac{3}{5\left[\frac{6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}}\right]^\frac{2}{3}}$ etc
• Aug 17th 2008, 03:14 PM
Kingreaper
Thanks, lol here are a couple more, and these are the last ones I need help with on the packet.

1. Forces of 85 pounds and 50 pounds act on a single point. The angle between the forces is 15 degrees. Find the dirrection and magnitude of the resultant force.

2. A pet supply company mixes 2 brands of dry dog food. Brand X costs $15 per bag and ciontains 8 units of nutritional element A, 1 unit of nutritional element B, and 2 units of nutritional element C. Brand y costs$30 per bag and contains 2 units of nutritional element A, 1 unit of nutritional element B, and 7 units of nutritional element C. Each bag of mixed dog food must contain at least 16 units of A, 5 units of B, and 20 units of C. Find the number of bags of brands X and Y that should be mixed to produce a mixture meeting the minimum nutritional requirements and having a minimum cost.

3. 2 planes start from the same airport and fly in opposite directions. The second plane starts 1/2 hour after the first, but travels 80 km faster. Find the air speed of each plane if 2 hours after the first plane departs the planes are 3200 km apart.

4.2 ships leave port and 9 am. one travels at a bearing of N 53 degrees W at 12 mph and the other travels at a bearing of S 67 degrees W at 16 mph. Approximate how fart apart they are at noon that day.

5. A 55 gallon barrel contains a mixture with a concentration of 30%. How much of this mixture must be withdrawn and replaced by a 100% concentrate to bring the mixture up to 50% concentration?

6. Pluto moves in an elliptical orbit with the sun at one of the foci. The lengh of half the major axis is 3.666 x 10^9 miles and the eccentricity is 0.248. Find the smallest and greatest distance of Pluto from the center of the sun.

7.A right circular cone has a base of radius 1 and a height of 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side length of the cube?

8. A worked can cover a parking lot with asphalt in 10 hours. With the help of an assisstant, the work can be done in 6 hours. How long would it take the assistant working alone to cover the parking lot?