Precalculus

Basic Functions

100

Find absolute maximum of the function f(x) = 25 - x^2

(0, 25)

200

The equation of a line in point-slope form that passes through the points (1,2) and (-3,4)

m=2/-4 = -1/2

y-2=-1/2 (x-1)

300

The composition f(g(h((x))) if f(x)=3x²-4, g(x)=2x+1, h(x)=1/x

3(2/x+1)²-4

400

The roots of f(x)=(x^2+4x+4)/(x-1)

x=2

500

The roots of the quadratic function f(x) = x^2 - 2x + 5

x=1+4i , x=1-4i

Domain & Range

100

The domain of (x+1)^1/2 in interval notation

[-1, inf)

200

The domain of (2x+1) / (x-5)(2x) in interval notation

x can't be zero or 5 but can be everything else

(-inf, 0) U (0,5) U (5, inf)

300

The range of y=5^x-3+1 and the equation(s) of any asymptote(s)

(1, inf) and horizontal asymptote is y=1

400

The domain, asymptote(s), and x-intercept of y=log₅(2x+3)-1

Domain: 2x+3>0, so x>-3/2

VA: x=-3/2

x-int: (1,0) because 0=log₅(2x+3)-1, so 1=log₅(2x+3), so 5¹=2x+3, which gives x=1

500

The domain and range of y=2 sin(2x-pi)-3

D: all real numbers

R: [-5,-1]

Polynomials & Rationals

100

The leading coefficient of f(x) = 5x³+ 3x⁸ - x⁴

3x⁸

200

The end behavior of f(x)=-3x⁷+5x⁴-4x³+9

Negative LC, odd degree so end behavior is like -x³

300

The result of 2x³-6x+1 divided by x-3

2x²+6x+12+ 37/(x-3)

400

The asymptote(s) of the rational function

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

x=1 and x=-2/3

500

The asymptote(s) of the rational function

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

VA: x=-2/3, x=1/2 and x=3

Exponential & Logs

100

The solution to 4^x=7

x=log₄7 because log and exponential functions of same base are inverses

200

The solution to 5^3x-1=5^x

x=1/2

300

The value of log₃(81⁵)

log₃((9²)⁵)

log₃(((3²)²)⁵)

log₃(3²⁰)

=20

400

The solution to log₇(x-2)+log₇(x+3)=log₇14

x=4

500

The solution of log₂(x-4)=3

x=4

Trig

100

The conversion of 160 degrees to radians

160/x = 180/pi 

x=160pi/180

=8pi/9

200

The amplitude and period of y=2sin(4x)

amplitude=2

frequency=4, so period=2pi/4 = pi/2

300

The value of tan^-1(-1)

The range for tan^-1(x) is (-pi/2, pi/2)

tan^-1(-1) means we need x so that sin x / cos x = -1, which means sine and cosine must be the same value at this angle, but opposite signs (Q4)

tan^-1(-1) = -pi/4

400

From a point on the ground 47 feet from the foot of a tree, the angle of elevation to the top of the tree is 35º. Find the height of the tree to the nearest foot. (Calculator allowed)

33

500

The solution to tan(x)=1

pi/4