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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 -...

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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 +...

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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 +...

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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 =...

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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 =>...

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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 =...

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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...

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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...

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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...

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

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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...

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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...

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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...

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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 =...

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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...

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The difference between solving |4x-4| =12 and 4x-4 =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...

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To test if 3 - i is a root of p=x^4-4x^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...

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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)...

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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...

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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 =...

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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 =...

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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...

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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...

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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...

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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...

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We have to find x and y given x(1+2i)+y(2-i)=4+3i x(1+2i)+y(2-i)=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...

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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 =...

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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)

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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...

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-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...

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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...

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

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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...

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The sum of n terms of an arithmetic series is given as: Sn = 5n^2 - 11n Sn-1 = 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(z-1) = -2 i(z-1) = -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...

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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...

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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...

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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)=1-2x f(f(x)) = f( 1- 2x) = 1 - 2(1-2x) = 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...