Exam 2 Review

Polynomials

100

Decide if the following are polynomials. If they are: state the degree. If they are not: say why not

f(x)= x²+3x⁵-7x³+ 𝝿 

g(x)=x³-21x²+3^x+10

f(x): Yes! degree=5

g(x): No! Polynomials don't have variables in the exponent

200

List all the possible rational roots of the following polynomial:

h(x) = 3x³ + 5x² - x + 8

+-1, +-2, +-4, +-8, +-1/3, +-2/3, +-4/3, +-8/3

300

Let g(x) = x³. List all of the transformations applied to g(x) in order to obtain f(x) = -(2x+6)³-2

First rewrite as f(x) = -(2(x+3))-2

Horizontal shrink by a factor of 1/2, horizontal shift left by 3, reflect over the x-axis, vertical shift down 2.

400

Let f(x) = (x+2)(x-4)(3x+4)(2x-2)

Find the following:

-all the x-intercepts

-the degree of the polynomial

-the limit as x approaches infinity 

-the limit as x approaches negative infinity

x-intercepts: -2, 4, -4/3, 1

degree = 4

both limits go towards infinity

500

Fully factor the following polynomial using the rational root theorem and polynomial long division.

p(x) = x³ - 2x² - 5x + 6

Possible rational roots: +-1, +-2, +-3, +-6

p(x) = (x-1)(x+2)(x-3)

Exponential Functions

100

Write the parent function for an exponential function with base 1/2 and graph it (label two points and any asymptotes)

f(x) = (1/2)^x

H.A. y=0

200

State the domain and range of 

g(x) = 5^x+2 - 7

Domain: all real numbers

Range: (-7, infinity)

300

Find the limit as x approaches infinity of f(x) and the limit as x approaches negative infinity of f(x) 

f(x) = (1/4)^-x + 1

limit towards infinity: infinity 

limit towards negative infinity: 1

400

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

State the parent function, graph the parent function, then graph g(x) using transformations.

Parent function: f(x) = (1/3)^x

500

Rewrite the following equation in the form f(x) = b^x and state the value of b:

f(x) = (1/5)^x(25)^2x(125)^x/3

f(x) = 625^x

b=625

Logarithmic Functions

100

Rewrite the following in logarithmic form

1331 = 11³

log₁₁(1331) = 3

200

Simplify the following to a single logarithmic expression

log₄(x) + log₄(2x+1) - 3log₄(y)

log₄((2x²+x)/y³)

300

State the domain and range of the following:

f(x) = log₉(3x+6) + 1

Domain: (-2, infinity)

Range: all real numbers

400

g(x) = log₃(x-1) + 2

State the parent function, graph the parent function, then graph g(x) using transformations.

Parent function: f(x) = log₃(x)

500

State the end-behaviors of the following logarithmic equation:

f(x) = log₃(-x+2) - 5

limit as x approaches 2 from the left is negative infinity

limit as x approaches negative infinity is infinity

Inverses

100

How do you know if a function has an inverse by looking at the graph?

A function has an inverse if it passes the horizontal line test

200

Find the inverse of 

f(x) = 3x + 12

f^-1(x) = (1/3)x - 4

300

Find the range of the inverse of

f(x) = 5/(x - 4)

Range of f^-1(x): (-infinity, 4) U (4, infinity) 

The range of f^-1(x) is the domain of f(x)

400

Let f(x) = x² + 3 

What is f^-1(19) ?

f^-1(19) = 4 because f(4) = 19

500

Find the inverse of 

f(x) = (2x+3)/(x-4)

f^-1(x) = (3+4x)/(x-2)

Solving Equations Exp/Log

100

Solve the following exponential equation for x:

3^x+9 = 3^7x-6

x= 5/2

200

Solve the following logarithmic equation for x:

log₇(2) + log₇(x) = log₇(3x-2)

x = 2

300

Solve the following exponential equation for x:

4^3x+2 = 8^x(4³)

x = 2/3

400

Solve the following logarithmic equation for x:

log₅(6) + log₅(2x²) = log₅(96) - log₅(2)

x = 2, -2

500

Solve the following logarithmic equation for x:

log₄(12+x) + log₄(x) = 3

x = 4 

x = -16 is not included because it is outside of the domain