MATH 1314 CH 3, 4

Polynomial Inequality

100

2^2-2x > 3

(-∞,-1) U (3, ∞)

200

2x^2- x > 5

(-∞,-3)U( 5/2 ,∞)

300

x^3+3x^2≤x+3

(-∞,-3]U[-1,1]

400

x^3+x^2≤4x+4

(-∞,-2]U[-1,2]

500

x^3≥9x^2

{0}U[9,∞)

Rational Inequality

100

(3x+3)/(2x-4)>0

(-∞,-1)U(2,∞)

200

(x-1)/(x+1)≥0

(-∞,-1)U[1,∞)

300

(4-2x)/(3x+4)≤0

(-∞, -4/3) U [2, ∞)

400

(x+1)/(x+3)<2

(-∞,-5) U (-3, ∞)

500

(x+4)/(2x-1)≤3

(-∞,1/2)U[7/5, ∞)

Equations

100

sqrt(20-8x)=x

x=2

200

sqrt(x+10)=x-2

x=6

300

5^(3/2)-25=0

.^3sqrt(25)

400

x^(2/3)-x^(1/3)-6

x= 27 and x=-8

500

7x^(2/3)-55x^(1/3)-8=0

x=512 and x=-1/343

Exponential function

100

Graph and determine its doman, range and asymptote

f(x)= (1/3)^x

D: (∞ ,-∞)

R: (0, ∞)

Asymptote: y=0

Decreasing if 

0<b<1

200

Graph and determine its doman, range and asymptote

f(x)= 3^(x-4)

D: (∞ ,-∞)

R: (0, ∞)

Asymptote: y=0

-Right 4 translation

Increasing if 

b >1

300

Graph and determine its doman, range and asymptote

f(x)= -3^(x+1)+2

D: (∞ ,-∞)

R: (2, -∞)

Asymptote: y=2 

-Relfection across x-axis

-2 up translation

-1 left translation

400

Slove the following exponential equations

1) 27^(x+3)=9^(x-1)

2)5^(3x-6)=125

1)x=-11

2)x=3

500

Solve

e^(2x)-4e^x+3=0

x=ln3, 0

Logarithmic functions

100

Expand the following expression as much as possible

log₂((sqrtx)/(36y^4))

1/3log₂x-2-4 log₂ y 

200

Write as a single logarithm 

 

1/2(log _5x+log_5y)-2log_5(x+1)

log_5((sqrtxy)/(x+1)^2)

300

 Solve

log_2x+log_2(x-7)=3

x=8

400

Solve

ln(x+2)-ln(4x+3)=ln(1/x)

Hint: apply the one-to-one property of logarithms 

x=3

500

Solve

log_2(x-6)+log_2(x-4)-log_2x=2

 

x=12