Exponentials and Logarithms

Composite Functions

100

f(x) = 2x+3

g(x) = x²

Solve for g(f(x))

(2x+3)²

200

f(x) = 3x²+5

g(x) = 2/x

Solve for g(f(x))

2/(3x²+5)

300

f(x) = x² - x - 2

g(x) = (x-2)/x

What is the domain of g(f(x))?

(-inf, -1) u (-1, 2) u (2,inf)

400

If f(x) = 5x + 1 and f(g(x)) = 10x + 16, what is g(x)?

g(x) = 2x + 3

500

f(x) = 2x + 9/x

g(x) = x² - 1

h(x) = sqrt(x)

What is h(g(f(x)))

sqrt((2x + 9/x)²-1)

Inverse Functions

100

Solve for the inverse of f(x) = 4x + 2

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

200

Solve for the inverse of f(x) = x³+1

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

300

Solve for the inverse of f(x) = mx + b

f^-1(x) = (x-b)/m

400

True/False: f(x) = x³-2x²-4x+8 has an inverse function

False. Does not pass y = 0 horizontal line test

500

Find the inverse of f(x) = x³-3x²+3x-10

f^-1(x)=(x+9)^(1/3) + 1

Exponentials

100

Solve for x: 

4^x - 12(2^x) + 32 = 0

x = 2, x = 3

200

Solve for x: 

16^x - 14(4^x) - 32 = 0

x=2

300

2^x has been vertically stretched by 3, horizontally flipped, and then shifted down by 4. What is the equation of the transformation?

f(x) = 3(0.5)^x - 4

400

Solve for x:

2^2x-1 - 3*2^x-1 + 1 = 0

x = 0, x = 1

500

Solve for x:

3^2x-5 - 28(3)^x-3 + 9 = 0

x=5, x=2

Log ID

100

Solve for x:

x = log₃81

4

200

Solve for x:

x = log_0.532

-5

300

Solve for x:

x = log_sqrt(2)64

12

400

Solve for x:

x = log_0.2532

-5/2

500

If a = log₁₀x and log₆36^a=6, what is x?

1000

Log Properties

100

Combine as a single logarithm:

3log_bX - log_bY - 3log_bZ + 0.5log_bW

log_b(X³W^1/2 / YZ³)

200

Expand. Write in terms of log_bX, log_bY, log_bZ.

log_b(XY)-logb(Y²Z)

log_bX - log_bY - log_bZ

300

Simplify

log₃64/log₃8

3

400

Combine as a single logarithm:

-2+3log_bX-0.25log_bY

log_b(X³ / b²Y^0.25)

500

Solve the following equation:

log₄(x-6)-log₄x=2

8