Pre-Calculus Benchmark Review

Functions

100

If f(x) = 10x - 3, what is f(u+v)?

f(u+v) = 10(u+v) - 3 = 10u + 10v - 3

200

If g(x) = x^2 + 4x + 3, what is g(3j)?

g(3j) = (3j)^2 + 4(3j) + 3 = 9j^2 + 12j + 3

300

If h(x) = (x+3) / (x^2 - x + 1), what is h(-kn)?

h(-kn) = (-kn + 3) / ((kn)^2 + kn + 1)

400

What is the difference quotient of f(x) = 2x^2 - 3?

(f(x+h) - f(x)) / h = 4x + h

500

Determine whether f(x) = x^4 + 3x^2 + 1 is even, odd, or neither.

Neither.

Polynomials

100

Find the vertex of h(x) = x^2 + 4x + 8.

(-2,4)

200

Place h(x) = x^2 + 2x - 3 in to standard form.

h(x) = (x+1)^2 - 4

300

Place g(x) = 3x^2 - 12x + 13 in to standard form.

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

400

Factor x^4 - 2x^3 + 2x - 1 completely.

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

500

Find all the zeros (real and complex) of x^4 + x^3 - x^2 + x - 2.

x = -2, 1, i, -i

Rational Functions

100

Where are the asymptotes of r(x) = (x+2) / (x-3)?

x = 3

200

What is the horizontal asymptote of r(x) = (x^3 - x + 1) / (x^4 + 2x^2 + 4)?

y = 0

300

Where are the vertical asymptotes of r(x) = (x^2 + x - 2) / (x^3 + 2x^2 - x - 2)?

x = 1, x = -1, x = -2

400

What is the horizontal asymptote of r(x) = (x^4 - 3x^3 + 3x - 6) / (3x^4 + x^3 - x^2 + 2)?

y = 1/3

500

Let r(x) = ((x+1)(x+2)(x+3)(x-4)(x-5)) / ((x-1)(x-2)(x-3)(x+4)(x+5)). Where are the x-intercepts and vertical asymptotes of r?

x-intercepts: x = -1, -2, -3, 4, 5 vertical asymptotes: x = 1, 2, 3, -4, -5

Exponents

100

Solve 2^x = 16.

x = 4

200

Solve e^x = 6.

x = ln 6

300

Solve 4^(3x+2) = 4^(2x).

x = -2

400

Solve e^(3x) = 3.

x = (ln 3) / 3

500

4 ( 2^(4x) ) = 12

x = (ln 3) / (4 ln 2)

Logarithms

100

Solve ln x = 4.

x = e^4

200

Solve log_2 x = 5.

x = 32

300

Solve ln (x+3) = ln (2x+1).

x = 2

400

Solve 3 log_3 (x+3) = 9.

x = 24

500

Solve ln (x+1) + ln (x-3) = ln (x-5).

x = 1, 2