Polynomials
What are the zeros and end behaviours of f(x)=(x+2)(x-5)?
Zeroes at x=-2 and x=5
End behaviours: as x-> negative infinity, f(x)-> negative infinity
as x-> positive infinity, f(x)-> positive infinity
Solve for x:
4=-9^x+7
x=1/2
log(3x^3)
Consider f(x)=(2x-6)/(4x+2), where are the vertical and horizontal asymptotes?
Vertical: x=-1/2
Horizontal: y=1/2
No calculators
Evaluate sin(pi/3)+cos(pi/6)+tan(pi/4)
1+sqrt(3)
What is the remainder when f(x)=x^3+6x-8 is divided by (x-1)?
-1
What is the inverse of f(x)=2(3^x)+1
y = log_3 [ (x-1)/2 ]
(log base 5 of 125^4)
12
What is the equation of the oblique asymptote in
f(x)=2x - 5 + 4/(x-3)
y=2x-5
What is the period of f(x)=2cos[pi/4(x+5)]-2
Determine the value of k such that (x+4) is a factor of 3x^3+11x^2-6x+k
k=-8
What is the range of f(x)=-4[2^(5x)]+7 ?
Range: {y|y<7, y is a real number}
Describe the transformations from y=log(x) to get f(x)=-3log(x/2+6)+10
vertical reflection, vertical stretch by a factor of 3, horizontal stretch by a factor of 2, horizontal shift by 12 to the left, vertical shift up by 10 units
What is the range of f(x)=(2x^2-18x+36)/(5x^2+10x-75)?
R: {y|y is not equal to -3/20 or 2/5, y is a real number}
What is the domain of f(x)=cot[pi/3 (x-1)]?
D: {x|x not equal to 1+3k, k is an integer, x is a real number}
Consider a polynomial with roots at x=2, -2, [5+sqrt(2i)]/3, [5-sqrt(2i)]/3. Will the leading coefficient be positive or negative if the y-intercept is at 108?
(can you sketch / find the equation of the function?)
Negative
Solve for x in 2^(3x+3)-2^(3x+1)=384
x=2
Solve log_7 (x+2) - log_7 (x-4) = -1
x=3
Solve the inequality 3/(x-5)(x+2) <= 0
x is an element of the interval (-2, 5)
Solve cos(x)cos(2x)-sin(x)sin(2x)=0 for all real x values
x=pi/6 + 2pi*k or pi/2 + 2pi*k, k is a real number
Solve the inequality -x^7+8x^6+2x^5-43x^4+300x^3-772x^2+912x-896 > 0
(hint: factor / graph!)
(negative infinity, 4) U (2,8)
x=1/7
What is the domain of f(x)=log [(4-x)(2-x)(1+x)]?
x is an element of (-1, 2) U (4, positive infinity)
Determine the coordinates of the holes in
f(x)=(3x^5-65x^4+545x^3-2215x^2+4372x-3360)/(4x^4-73x^3+476x^2-1307x+1260)
Holes at (4, 4/7), (5, 14/11), (7, 52/19)
Prove the trig identity:
sin(2x)+cot(x)sin(x)(sinx+cosx) = (3cosx-sinx) / csc(x) + 1
Solutions may vary.