# Thread: Algebra 2 problems, need help

1. ## 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

2. Hello, Kingreaper!

Do you know how to find an inverse function?

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

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

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

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

(3) Solve for $y$

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

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

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

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

Note that this inverse is not truly a function.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3. 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))

4. 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 ......

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

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

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.

5. yes, those are the equations, lol. Thanks for the latex tip, I'm gonna try that now.

6. 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 ....

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

and

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

Also;

$fg(x) = f\left[\frac {3}{5x^\frac{2}{3}}\right]
\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

$gf(x) = g\left[\frac {6x^\frac{1}{2} + x^2}{(x-2)^\frac{1}{5}} \right]
etc

7. 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?