0. Functions

Find the inverse of the function ƒ(x) = 2x. Verify that ƒ(ƒ⁻¹(x)) = x and ƒ⁻¹(ƒ(x)) = x .

Sketch the graph of the inverse of ƒ. <IMAGE>

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

ƒ(x) = |2x + 1|

Where do inverses exist? Use analytical and/or graphical methods to determine the largest possible sets of points on which the following functions have an inverse.

{Use of Tech} ƒ(x) = 1/(x-5)

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = x² -2x + 6

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = (x + 1)³

Finding all inverses Find all the inverses associated with the following functions, and state their domains.

ƒ(x) = 2 / ( x² + 2)

Solving equations Solve the following equations.

log₁₀ x= 3

Solving equations Solve the following equations.

ln x= -1

Solving equations Solve the following equations.

7ˣ = 21

Solving equations Solve the following equations.

3(ˣ³⁻⁴) = 15

Solving equations Solve the following equations.

5(ˣ³) = 29

Solving equations Solve each equation.

ln 3x + ln (x + 2) = 0

The parabola y=x²+1 consists of two one-to-one functions, g₁(x) and g₂(x). Complete each exercise and confirm that your answers are consistent with the graphs displayed in the figure. <IMAGE>

Find formulas for g₁((x) and g₁⁻¹(x). State the domain and range of each function.

Determine whether the following statements are true and give an explanation or counterexample.

(4x+1)^{ln x} = x^ln(4x+1)