Combinations of Functions

Read this section for an introduction to combinations of functions, then work through practice problems 1-9.

Practice Problem Answers

Practice 1: $C(x)$ is the cost for one night on date $x$ .

$C(x) = \left\{ \begin{array} {ll} $50 & \text { x is between June 1 and September 15} \\ $40 & \text { if x is any other date } \end{array} \right.$

Practice 2: See Fig. 29

$x$	$g(x)$
-3	-3
-1	2
0	2
1/2	2
1	undefined

$x$	$g(x)$
π/3	-π/3
2	-2
3	-3
4	undefined
5	1

Practice 3:

$f(x)=\left\{ \begin{array} {lll} 1 & \text { if } x ≤ -1 \\1 -x & \text { if } -1 ≤ x ≤ 1 \\ 2 & \text { if } 1 < x \end{array} \right.$

Practice 4:

$f \circ g(3) = f(2) = 2/–1 = –2$	$f \circ g(8) = f(3)$ is undefined	$g \circ f(4) = g(4) = \sqrt 5$
$f \circ h(1) = f(2) = 2/–1 = –2$	$f \circ h(3) = f(2) = –2$	$f \circ h(2) = f(3)$ is undefined
$h \circ g(–1) = h(0) = 0$	$f \circ g(x) = f(\sqrt{1 + x} ) = ( \sqrt{1+x} )/( \sqrt{1+x} – 3)$ , $g \circ f(x) = g(\dfrac{x}{x–3} ) =\sqrt{1 +\dfrac{x}{x–3}}$

Practice 5: See Fig. 30.

Practice 6: $f(x) = \dfrac{9/x + x}{2}$ .

$f(1) = \dfrac{9/1 + 1}{2} = 5$ , $f(5) = \dfrac{9/5 + 5}{2} = 3.4$ , $f(3.4) ≈ 3.023529412$ ,

$f(3.023529412) ≈ 3.000091554$ , and $f(3.000091554) ≈ 3.000000001$ .

These values are approaching 3, the square root of 9.

Putting $A = 6$ , then $f(x) = \dfrac{6/x + x}{2}$ .

$f(1) = \dfrac{6/1 + 1}{2} = 3.5$ , $f(3.5) = \dfrac{6/3.5 + 3.5}{2} = 2.607142857$ ,

$f(2.607142857) ≈ 2.45425636, f(2.45425636) ≈ 2.449494372$ ,

$f(2.449494372) ≈ 2.449489743$ .

$f(2.449489743) ≈ 2.449489743$ (the output is the same as the input for 9 decimal places)

These values are approaching 2.449489743, the square root of 6.

For any positive value $A$ , the iterates of $f(x) =\dfrac{A/x+x}{2}$ (starting with any positive $x$ ) will approach $A$ .

Practice 7: Fig. 31 shows some of the intermediate steps and final graphs.

Practice 8: Fig. 32 shows the graph of $y = x2$ and the graph (thicker) of $y = INT( x^2 )$ .

Practice 9: Fig. 33 shows the "holey" graph of $y = x$ with a hole at each rational value of $x$ and the "holey" graph of $y = sin(x)$ with a hole at each irrational value of $x$ . Together they form the graph of $r(x)$ .

(This is a very crude image since we can't really see the individual holes which have zero width).