0. Functions

Splitting up curves The unit circle x² + y² = 1  consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x)  (see figure) <IMAGE>.

a. Find the domain and a formula for each function.

Splitting up curves The unit circle x² + y² = 1  consists of four one-to-one functions, ƒ₁ (x), ƒ₂(x) , ƒ₃(x), and ƒ₄ (x)  (see figure)<IMAGE>.

b. Find the inverse of each function and write it as y= ƒ⁻¹ (x)

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f⁻¹ (10)

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f⁻¹( g⁻¹(4))

Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. <IMAGE>

f^-1(1 + f(-3))

A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs \$5/ft for the legs and \$10/ft for the hypotenuse, write the total cost C of construction as a function of h.

Theory and Examples

The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10.

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

sin x²

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

h(t) = |t³|

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x/(x² − 1)

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

g(x) = x⁴ + 3x² − 1

Even and Odd Functions

In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.

f(x) = x² + x

Increasing and Decreasing Functions

Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

y = −x³

The Greatest and Least Integer Functions

Does ⌊x⌋ = ⌈x⌉ for all real x? Give reasons for your answer.

The Greatest and Least Integer Functions

For what values of x is

b. ⌈x⌉ = 0