0. Functions

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)

Explain why b^x = e^xlnb.

Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.

b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.