1. Find f(g(h(x))) where x=4x^4+3x, h=19x^6+12x^4+3, g=19x^7+4x^6-22x^5+44x^4
To start, you first need to find h(x). In order to do this, substitute the variables in the function h with the entire function x. This will give you 19(4x+3x)^6+12(4x+3)^4+3. After that is done, distribute the factors out of the equation and Distributed out, this equation should simplify to (2235331 x^6+84 x+3). Now, to find f(g(h(x)), you must substitute the variables in the function f(g) with the entire function h(x). The setup for this problem will be 19 (19 (4 x+3 x)^6+12 (4 x+3 x)+3)^7+4 (19 (4 x+3 x)^6+12 (4 x+3 x)+3)^6-22 (19 (4 x+3 x)^6+12 (4 x+3 x)+3)^5+44 (19 (4 x+3 x)^6+12 (4 x+3 x)+3)^4. Now you once again distribute all of the factors and combine like terms. This should make the equation f(g(h(x)))= 5298421303499822062831772727828348207383605009x^42+1393740670378523526468823795654007726793732 x^37+50275464568792067773624775299728348898843x^36 +157123329357212837235355120340034628944 x^32+11335607184202787935167940117621545912 x^31+203202891792126269604683474970755291 x^30+9840719835232364787563773261143360 x^27+1064932893017895544948496408227920 x^26+38180123906791894906824564007620 x^25+454625152835284402740199896689 x^24+369797791092021111036959159040 x^22+53357862445429469297491779840 x^21+2869490387034867920834331360 x^20+68336211215545062149948784 x^19+609541601397062395346943 x^18+8337829462857117803252736 x^17+1503824415293330860504320 x^16+107830649022864580391040 x^15+3851940769916704877664 x^14+68716661448375978156 x^13+104931502806140818101 x^12+22604482447501473792 x^11+2026047712379201280 x^10+96499660728248064 x^9+2582257196658384 x^8+597569158551960 x^7+141793579673109 x^6+15227096822784 x^5+906573959424 x^4+32345635392 x^3+693273168 x^2+8285004 x+42687
2. Find the domain of f(x)= sqrt(104x^4+1150x^3-6462x^2-28872x+22680), x=-14, x=6
In order to start this problem, we need to find a way to reduce this polynomial into a factorable equation. Since we are given several x intercepts, we know that we must use long division to create a factorable polynomial. We will use (x+14) for our first x intercept to long divide. Also, we can factor out any greatest common factors in the equation, which in this case will make the equation 2(52x^4+575x^3-3231x^2-14436x+11340). So, our long division problem should look like this: (x+14)/(52x^4+575x^3-3231x^2-14436x+11340). The first thing we must do to this problem is divide the first part of the divisor by the first term, which will give us 52x^3 as part of the dividend. Next, we distribute the divisor (x+14) into the term of the divisor we just found, giving us 52x^4+728x. This value is now subtracted from the two terms above it, which will give us (52x^4+575x^3)-(52x^4+728x^3). This will give us -153x^3, which we will use as the next term of our dividend. Then we drop the next term down next to the difference we found in the last subtraction problem, and repeat those steps with this next term. Do this with every term until you end up getting a difference of 0. The final divisor of this problem should be 52x^3-153x^3-1089x+810. Once this is done, we have to find out if this new polynomial is factorable. Try the grouping method if your polynomial has four terms, and if this method does not work, we must use another x intercept to further reduce the polynomial into a factorable equation. So, we repeat the steps above for long division, this time using (x-6) as the divisor. This long division should yield the polynomial 52x^2+159x-135. Now that we have found a factorable quadratic equation, we use basic factoring to find the remaining x intercepts. Factoring this equation will give us the factors (13x-9)(4x+15). To find the x intercepts from this, we must find where the factors are equal to zero. In this case, that will be x=9/13 and x=-15/4. This means we know our x interceps, which are x=-14, x=6, x=9/13, and x=-15/4. Using these x intercepts, create a graph. The domain of the function will be all values where x>0. This makes the domain (-inf, -14]U[-15/4,9/13]U[6,inf).
3.Find the inverse of f(x)= (16x+9)/(15x+33)
To begin this problem, we first have to switch the variables from one side to the other, so the equation becomes x=(16y+9)/(15y+33). The very first step to finding the inverse of any radical function is to multiply both sides by the denominator. This means we distribute x(15y+33), making our equation 15xy+33x=16y+9. Now, we have to get all x values on one side, and all y values on the other side. Our objective is to make one of this side just y. To do this, we will subtract 33x from both sides, and subtract 16y from both sides. This leaves us with 15xy-16y=-33x+9. Finally, we divide out 15x-16, to leave y alone on the right side. This leaves us with y=(-33x+9)/(15x-16). The very last thing we need to do is completely factor the numerator, and we will do so by taking a greatest common factor of 3, leaving us with the final inverse equation, y=-3(11-3)/(15x-16)
4.Find the domain of f(x)= sqrt(8x^3+28x^2-16x-56)
In order to factor any four term cubic polynomial, the best method to use is the grouping method. We can take the equation out of the radical and set it equal to zero in order to find the x intercepts. After that, the first step to this method is to take out a greatest common factor from your equation, which in this case is 4. This makes our equation 4(2x^3+7x^2-4x-14). Now, we group the first and second terms together, and subtract the quantity of the third and fourth terms. For this example, it should now look like 4[(2x^3+7x^2)-(4x+14)] (note that in the second set, we want to switch the sign) of both terms). Now, we once again find the greatest common factor of both factor pairs, which will leave us with 4[x^2(2x+7)-2(2x+7)]. Now that we have a factor of (2x+7), which has no common factors, we take the greatest common factors of the two groups, giving us the second factor pair. This leaves us with 4(2x+7)(x^2-2). Now, we find where both of these factor pairs are equal to zero, giving us our x intercepts. Our x intercepts will be x=-7/2, x=-sqrt(2), and x=sqrt(2). Now we use the x intercepts to graph the equation. Our domain is going to be the area which the graph is at x>0. This makes our domain [-7/2, -sqrt(2)]U[sqrt(2, inf).
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