Unit 6 Review

Exponential Functions

100

True or False:

Exponential functions are continuous arithmetic sequences

False: Exponential functions are similar to geometric sequences, not arithmetic sequences.

200

True or False:

The function pattern associated with exponential functions is sum-product

True

300

simplify the following expression:

(a)^m(a)ⁿ

(a)^m+n

400

Given the following geometric sequence, find the common ratio:

16, -8, 4, -2, 1

r = -1/2

500

Given a common ratio of 3 and an initial value of 2, write the exponential function.

f(x) = 2(3)^x

Logarithmic Functions

100

True or False:

Logarithmic functions are the inverse of exponential functions.

True

200

True or False:

The function pattern associated with logarithmic functions is sum-product

False: Sum-product is the function pattern for exponential functions. Logarithms have the opposite function pattern: product-sum

300

Solve the following expression:

log_a(1) = 

0

400

Solve the following expression:

log₂(8) = ____

3

500

Find the value of x:

log₁₆(x) = 1/4

x = 2

Inverse of functions

100

True or False:

f^-1(f(x)) = x

True: Inverse functions cancel each other out.

200

True or False:

log_a(a^x) = x

True: Logarithms and exponentials are inverse of each other, so when we compose them they cancel each other out.

300

Rewrite the following equation in logarithmic form:

e^x = 5

ln(5) = x

Steps:

ln(e^x) = ln(5)

x = ln(5)

400

given the function y = 3x + 4, find the inverse:

y = (x - 4)/3

Steps:

switch x and y

x = 3y + 4

solve for y

x - 4 = 3y

(x - 4)/3 = y

500

Is the function f(x) = x² invertible? Justify.

No, because the function is not 1-1. For example, for f(x) = 9, x = 3 and x = -3. For a function to be 1-1 and thus invertible, each output can only have one corresponding input.

Function Composition

100

True or False: 

f(g(x)) = f o g(x)

True: these are just two ways to write the statement "f of g of x"

200

True or False:

For two functions f(x) and g(x) that are not inverses of each other, 

f(g(x)) = g(f(x))

False:

While this can occasionally be true, it is not necessarily true.

300

given the functions 

f(x) = 4 - 2x, g(x) = x²

find f(g(x))

f(g(x)) = 4 - 2x²

400

given the functions 

f(x) = 4 - 2x, g(x) = x²

find g(f(3))

g(f(3)) = -14

500

given the functions 

f(x) = 4 - 2x, g(x) = x²

find g(f(x))

g(f(x)) = 4x² - 16x + 16

Challenge

100

Given a function f(x) and its inverse g(x), solve the following expression:

g(f(3)) = 

g(f(3)) = 3

200

For the following expression, solve for x:

log(x+4) = 1

x = 6

Steps:

10^log(x+4) = 10¹

x + 4 = 10

x = 6

300

Carbon-14 is a radioactive element that decays such that half of the remaining sample remains after 5700 years. The relationship between the amount of carbon-14 remaining and the time that has elapsed since it started decaying can be represented by a ________ function. Fill in the blank.

exponential

400

Given f(x) = x² + 3x - 7 and g(x) = x + 2, Find f(g(x)).

f(g(x)) = x² + 7x + 3

500

Find the positive value of x that satisfies the following relation.

log₂(x) + log₂(x - 3) = 2

x = 4