
applications
fof^1(x) = f(f^1(x)) = x, by definition Here y = x/4 + 3 The inverse function is found by expressing y in terms of x, x = (y  3)*4. Now interchange y and x. f^(x) = (x  3)*4 fof^1(x) = f((x ...

applications
We have to find the square of (4+3i) (4+3i)^2 => (4+3i)(4+3i) => 4*4 + 4*3*i + 4*3*i + 3*i*3*i => 16 + 12i + 12i + 9i^2 i^2 = 1 => 16 + 24i  9 => 7 + 24i The required square is 7 +...

applications
To solve this, you need to split your first equation into two equations. This is because 3x+6 can equal 3x + 6 or it can equal (3x+6) which is 3x  6. So you need to solve two equations: 3x +...

applications
We have to find the tangent to the curve y = x^3  7x^2 + 14x  8 at the point where x=1 At x = 1, y = 1  7 + 14  8 = 0. The slope of the tangent to the curve at x = 1 is the value of y' at x =...

applications
For the triangle ABC AB = 6, B=pi/4, C=pi/6. As the angles of a triangle have a sum of pi. A = 7*pi/12 Use the property sin A/a = sin B/b = sin C/c c = 6, C = pi/6, B = pi/4 and A = 7*pi/12 =>...

applications
It is given that x = 5*cos t and y = 3*sin t. We have to find the equation of the tangent if t = pi/4 When t = pi/4 , x = 5*(1/sqrt 2) and y = 3/sqrt 2 dx/dt = 5*sin t and dy/dt = 3*cos t dy/dx =...

applications
If you know that tan x = (m/n), you can find sin x. Go about it this way: tan x = (m/n) => sin x/ cos x = m/n now change cos x to sin x, use (cos x)^2 = 1  (sin x)^2 => sin x/ sqrt( 1 (sin...

applications
The equations to be solved are: x^2 + y^2 = 200  xy ...(1) x + y = 20  (xy)^(1/2) ...(2) (2) => (x + y)^2 = (20  (xy)^(1/2))^2 => x^2 + y^2 + 2xy = 400 + xy  40(xy)^(1/2) => x^2 + y^2...

applications
Another method to solve x^2 + 3x – 4 = 0 is by factorization. x^2 + 3x – 4 = 0 => x^2 + 4x  x  4 = 0 => x( x + 4)  1(x + 4) = 0 => (x  1)(x + 4) = 0 => x = 1 and x = 4 The solution...

applications
We have to solve 3 log 4 (a2) = (9/2) for a 3 log(4) (a2) = (9/2) => log(4) (a2) = (9/6) => a  2 = 4^(3/2) => a  2 = 8 => a = 10 The solution for the equation is a = 10

applications
We need to find a if 729^a  57 = 0 729^a  57 = 0 => 729^a = 57 As the base cannot be equated we can use logarithms log (729^a) = log 57 use the property log a^b = b*log a => a*log 729 = log...

applications
Let us take the functions f(x) = x + 1 and g(x) = 3x to illustrate how to find f(g(x)) when f(x) and g(x) are given. f(g(x)) as g(x) = 3x => f(3x) use f(x) = x + 1 => 3x + 1 For x = c = 2...

applications
It is not possible to obtain the equation of a circle that passes through 2 points because there can be an unlimited number of circles that can pass through the two points. Like we need two points...

applications
The consecutive terms of a geometric sequence have a common ratio. We have the sequence given by 3 ,a ,b , 24 24/b = b/a = a/3 => 24a = b^2 or a = b^2/24 substitute in b/a = a/3 => 24b/b^2 =...

applications
The sum of n terms of a geometric sequence is given as a*(r^n  1)/(r  1), where r is the common ratio and a is the first term 4^n  1 => (2^(2n)  1) / (2  1) Here a = 1 and r = 2. But 2n can...

applications
The difference between solving 4x4 =12 and 4x4 =12 is that in the first case 4x  4 = 12 means (4x  4) = 12 and 4x  4 = 12 => 4x = 16 and 4x = 8 => x = 4 and x = 2 In the second...

applications
To test if 3  i is a root of p=x^44x^2+16, substitute x = sqrt 3  i x^4  4x^2 + 16 =>(sqrt 3 i)^4 4(sqrt 3 i)^2 + 16 => (sqrt 3  i)^2^2  4*(3 + i^2  2*i*sqrt 3) + 16 => (3 + i^2...

applications
We have to find f(3) given that f(3x) + 3*f(3x) = x + 1 f(3x) + 3*f(3x) = x + 1 for x = 1 => f(3) + 3*f(3) = 1 + 1 = 2 ...(1) for x = 1 => f(3) + 3*f(3) = 0 ...(2) (1)  3*(2) => f(3)...

applications
A polynomial that has roots 2 and 3 has factors equal to x  2 and x  3. This makes the polynomial equal to (x  2)(x  3) Opening the brackets and multiplying => x^2  2x  3x + 6 Adding...

applications
We have Int[ 4x^4 + 6x dx] = 10 => 4*x^4 / 4 + 6x^2 / 2 = 10 => x^4 + 3x^2 = 10 Let y = x^2 => y^2 + 3y  10 = 0 y1 = 3/2 + sqrt (49)/2 => y1 = 3/2 + 7/2 => y1 = 2 y2 = 3/2 ...

applications
The equation given is lg(x+1)  lg 9 = 1  lg x I assume the base of the logarithm is 10. Use the property that lg a  lg b = lg a/b and 1 = lg 10 lg(x+1)  lg 9 = 1  lg x => lg(x+1)  lg 9 =...

applications
We have y = 2x^3  3x^2. The solution of dy/dx = 12 is the first term of the arithmetic sequence with common difference 4. dy/dx = 12 => 6x^2  6x = 12 => x^2  x = 2 => x^2  2x + x  2 =...

applications
We have to solve x + y = 5 ...(1) x^2 + y^2 = 13 ...(2) From (1) x = 5  y Substituting in (2) (5  y)^2 + y^2 = 13 => 25 + y^2  10y + y^2 = 13 => 2y^2  10y + 12 = 0 => y^2  5y + 6 = 0...

applications
The polynomial that has roots 2 + 3i and 2 − 3i is given by (x  2  3i)*(x  2 + 3i) => (x  2)^2  (3i)^2 => x^2  4x + 4 + 9 => x^2  4x + 13

applications
To identify the multiplicity of a root, we have to find all the roots of the polynomial and also see if there are any roots which are equal. If a polynomial of order n has less than n roots, one or...

applications
We have to prove that (cospi/4+i*sinpi/4)^2008 is real. cos (pi/4) = 1/sqrt 2 sin (pi/4) = 1/sqrt 2 (cos pi/4 + i*sin pi/4)^2008 => [1/sqrt 2 + i*(1/sqrt 2)]^2008 => [1/sqrt 2 + i*(1/sqrt...

applications
An expression that can be represented in the form (x  a)^2 + (y b)^2 = r^2 is the equation of a circle with center at (a, b) and radius r. It has to be of the form given and not just one that has...

applications
We have to find x and y given x(1+2i)+y(2i)=4+3i x(1+2i)+y(2i)=4+3i => x + 2xi + 2y  yi = 4 + 3i equate the real and imaginary coefficients x + 2y = 4 and 2x  y = 3 x = 4  2y substitute in...

applications
This is like having two equations. One is 2x  3 = 7 and the other is (2x  3) = 7 Which is the same as 2x + 3 = 7 So now we have to solve both equations. 2x  3 = 7 2x = 10 x = 5 And 2x + 3 =...

applications
We have to solve for t given that t^8 = 8^2 t^8 = 8^2 => t^8 = 2^3^2 => t^8 = 2^6 => t = 2^(6/8) => t = 2^(3/4) The required value of t = 2^(3/4)

applications
To find the value of the integral of f(x)=(x^2+x), between the limits 1 and 3, we find the integral Int[ f(x) dx ], x = 1 to x = 3 => (x^3/3 + x^2/2), x = 1 to x = 3 => 3^3/3 + 3^2/2  1^3/3...

applications
2^94/4 is not equal to 2^92 + 4*2^90. Instead, 2^92 + 4*2^90 = 2^94/2^2 + 4*2^94 /2^4 => 2^94 / 4 + 2^94 / 4 => 2*2^94/4 => 2^94/2 Therefore we have 2^92 + 4*2^90 = 2^94 / 2 = 2^93

applications
Let the length of the rectangle be L. Its length is 2 more than the width. So width = L  2 The area of the rectangle is L*(L  2) When the dimensions are increased by 2, the area is increased by...

applications
We have to prove that arc sin x + arc cos x = pi/2 The left hand side is arc sin x + arc cos x take the sine of the angle sin (arc sin x + arc cos x) => sin (arc sin x)* cos (arc cos x) + cos...

applications
We have to solve x^2 +2x+5=0 by completion of squares. We use the relation x^2 + 2xy + y^2 = (x + y)^2 x^2 +2x+5=0 => x^2 + 2x*1 + 1^2 + 4 = 0 => (x + 1)^2 + 4 = 0 => (x + 1)^2 = 4 =>...

applications
To find the derivative dy/dx of y= x^25xy+3y^2 implicit differentiation can be used and differentiation carried out in the following way: y= x^25xy+3y^2 dy/dx = 2x  5*x*(dy/dx)  5y +...

applications
It is given that log(72) 48 = a and log(6) 24 = b a = log(72) 48 = log(6) 48/ log(6) 72 => log(6) (6*8)/log(6) 6*12 => [1 + log(6) 8]/[1 + log(6) 12] => [1 + 3*log(6) 2]/[2 + log(6) 2] b =...

applications
A complex number has to be found that has a square of 3i. Let the number be (a + ib) (a + ib)^2 = 3i => a^2 + 2iab + i^2*b^2 => a^2  b^2 + 2ab*i equate the real and imaginary coefficients...

applications
The sum of n terms of an arithmetic series is given as: Sn = 5n^2  11n Sn1 = 5(n  1)^2  11(n  1) => 5(n^2 + 1  2n)  11n + 11 => 5n^2 + 5  10n  11n + 11 => 5n^2  21n + 16 The term...

applications
We need the absolute value of x given that i(z1) = 2 i(z1) = 2 => z  1 = 2/i => z = 2/i + 1 => z = 2*i/i^2 + 1 => z = 2i / 1 + 1 => z = 1 + 2i The absolute value of z is...

applications
We have to solve 1/1024 = 4*16^2x for x. 1/1024 = 4*16^2x => 1/(2^10) = (2^2)*(2^4^2x) => 1/(2^10) = (2^2)*(2^8x) => [1/(2^10)]/(2^2) = (2^8x) => 2^(10  2) = (2^2)*(2^8x) => 2^12...

applications
The y intercept is the value of y when x = 0. When x = 0, the equation looks like this 15y = 120 Divide both sides by 15 and you get y = 8  that's the y intercept. The x intercept is the...

applications
All the powers of the imaginary part can be determined if we keep in mind the property that i is the square root of 1 or i^2 = 1. Higher powers are easily resolved once this is done. For...

applications
Take the sum of the bases. Multiply the sum of the bases by the height. Take that product and divide it by 2. That gives you the area of the trapezoid. In this case 15 + 20 = 35. 35*4 = 140...

applications
You want to solve 4 tan x  1 =  3*tan(x) Now tan x =  tan x 4 tan x  1 =  3*tan(x) => 4*tan x  1 = 3* tan x => tan x = 1 => x = arc tan 1 => x = pi/4 + n*pi and 5*pi/4 For the...

applications
The bacteria have an exponential growth rate of 70%. The formula for the exponential growth rate can be given by x (t) = a*b^ (t/T), where a is the initial number of bacteria, b is the increase in...

applications
Dependent equations are a set of equations where all the coefficients have a common multiple. This gets eliminated and you are left with a single equation. Attempting to solve a system of dependent...

applications
You know that f(x)=12x f(f(x)) = f( 1 2x) = 1  2(12x) = 1  2 + 4x = 1 + 4x f(f(1))+..+f(f(5)) => 1 + 4*1  1 + 4*2...1 + 4*5 => 1*5 + 4( 1+ 2+3+4+5) => 5 + 4*15 => 5 + 60...

applications
The equation of a line passing through two points (x1, y1) and(x2, y2) is (x  x1)/(y  y1) = (x2  x1)/(y2  y1) The points we have here are (2,3) and (5,8). Substituting the given values, we get...

applications
We have to prove: (2x1)^2(x3)^2 = (3x4)(x+2) Take the left hand side (2x1)^2(x3)^2 Use the relation x^2  y^2 = (x  y)(x + y) => (2x  1  x + 3)(2x  1 + x  3) => (x + 2)(3x  4)...