Chapter 2

The surface area of a sphere is given by \(A(r)=4 \pi r^{2},\) where \(r\) is in inches and \(A(r)\) is in square inches. The function \(C(x)=6.4516 x\) takes \(x\) square inches as input and outputs the equivalent result in square centimeters. Find \((C \circ A)(r)\) and explain what it represents.

The perimeter of a square is \(P(s)=4 s\) where \(s\) is the length of a side in inches. The function \(C(x)=2.54 x\) takes \(x\) inches as input and outputs the equivalent result in centimeters. Find \((C \circ P)(s)\) and explain what it represents.

Is it true that \((f g)(x)\) is the same as \((f \circ g)(x)\) for any functions \(f\) and \(g ?\) Explain.

Give an example to show that \((f \circ g)(x) \neq(g \circ f)(x)\).

Let \(f(x)=a x+b\) and \(g(x)=c x+d,\) where \(a, b, c,\) and \(d\) are constants. Show that \((f+g)(x)\) and \((f-g)(x)\) also represent linear functions.

Find \(\frac{f(x+h)-f(x)}{h}, h \neq 0,\) for \(f(x)=a x+b,\) where \(a\) and \(b\) are constants.