91Ó°ÊÓ

Even, Odd, or Neither? If \(f\) is an even function, determine whether \(g\) is even, odd, or neither. Explain. (a) \(g(x)=-f(x)\) (b) \(g(x)=f(-x)\) (c) \(g(x)=f(x)-2\) (d) \(g(x)=f(x-2)\)

Prove that if \(f\) is a one-to-one odd function, then \(f^{-1}\) is an odd function.

The length and width of a rectangular garden are 15 meters and 10 meters, respectively. A walkway of width \(x\) surrounds the garden. (a) Draw a diagram that gives a visual representation of the problem. (b) Write the equation for the perimeter \(y\) of the walkway in terms of \(x\) (c) Use a graphing utility to graph the equation for the perimeter. (d) Determine the slope of the graph in part (c). For each additional one- meter increase in the width of the walkway, determine the increase in its perimeter.

Restrict the domain of \(f(x)=x^{2}+1\) to \(x \geq 0 .\) Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

Right Triangle Explain how you could use slope to show that the points \(A(-1,5), B(3,7),\) and \(C(5,3)\) are the vertices of a right triangle.

It Is it possible for two lines with positive slopes to be perpendicular? Explain.

Use a graphing utility to compare the slopes of the lines \(y=m x,\) where \(m=0.5,1,2,\) and \(4 .\) Which line rises most quickly? Now, let \(m=-0.5,-1,-2,\) and \(-4 .\) Which line falls most quickly? Use a square setting to obtain a true geometric perspective. What can you conclude about the slope and the "rate" at which the line rises or falls?