91Ó°ÊÓ

Graph the function $$f(x)=\left\\{\begin{array}{ll}\lfloor x\rfloor, & x \geq 0 \\\| x\rceil, & x<0 \end{array}\right.$$ Why is \(f(x)\) called the integer part of \(x ?\)

Each gives a formula for a function \(y=f(x) .\) In each case, find \(f^{-1}(x)\) and identify the domain and range of \(f^{-1} .\) As a check, show that \(f\left(f^{-1}(x)\right)=f^{-1}(f(x))=x\). $$f(x)=x^{2}-2 b x, \quad b>0 \text { and constant, } x \leq b$$

a. Find the inverse of the function \(f(x)=m x,\) where \(m\) is a constant different from zero. b. What can you conclude about the inverse of a function \(y=f(x)\) whose graph is a line through the origin with a nonzero slope \(m ?\)

What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$y=-x^{3}$$

What happens if you take \(B=A\) in the trigonometric identity \(\cos (A-B)=\cos A \cos B+\sin A \sin B ?\) Does the result agree with something you already know?

Graph the functions. $$y=|x-2|$$

Graph the functions. $$y=|1-x|-1$$

Show that the graph of the inverse of \(f(x)=m x+b,\) where \(m\) and \(b\) are constants and \(m \neq 0,\) is a line with slope \(1 / m\) and \(y\) -intercept \(-b / m\)

What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$y=-\frac{1}{x^{2}}$$

What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing. $$y=-\frac{1}{x}$$