0. Functions

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\(\left\)(x\(\right\))=2+x$

The Greatest and Least Integer Functions

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

Solving trigonometric equations Solve the following equations.

cos²Θ = 1/2 , 0 ≤ Θ < 2π

{Use of Tech} Sum of squared integers Let T (n) = 1² + 2² + ... + n², where n is a positive integer. It can be shown that T (n) = n (n + 1) (2n + 1) / 8

c. What is the least value of n for which T(n) > 1000?

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

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\(\left\)(x\(\right\))=4x^3+\(\frac\)12x^{-1}-2x+1$

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.

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=\(\ln\]\left\)(3x+1\(\right\))$

Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.

cot (-17π/3)

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.

e. At what time is the ball at a height of 10 ft on the way down? 

Find the inverse function (on the given interval, if specified) and graph both $f$ and $f^{-1}$ on the same set of axes. Check your work by looking for the required symmetry in the graphs.

$f\(\left\)(x\(\right\))=x^2+4$, for $x\(\geq{0}\)$

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 at the origin

a. If ƒ(0) is defined and ƒ is an even function, is it necessarily true that ƒ(0) = 0? Explain.

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))

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)