Functions and Their Graphs

Practice Problem Answers

Practice 1:

Input	Output
$x$	$\dfrac{3}{x}+1$
5	$\dfrac{3}{5}+1 = 1.6$
$a$	$\dfrac{3}{a}+1$
0	$g(0) =\dfrac{3}{0} +1$ which is not defined because of division by 0.

Input	Output
$c+3$	$\dfrac{3}{c+3} +1$
$x+h$	$\dfrac{3}{x+h} +1$

Practice 2: $g(t) = t^2 – 5t$ .
$g(1) = 1^2 – 5(1) = –4$ . $g(–2) = (–2)^2 – 5(–2) = 14$ .
$g(w + 3) = (w + 3)^2 –5(w + 3) = w^2 + 6w + 9 – 5w – 15 = w^2 + w – 6$ .
$g(x + h) = (x + h) – 5(x + h) = x^2 + 2xh + h^2 – 5x – 5h$ .
$g(x + h) – g(x) = ( x^2 + 2xh + h^2 – 5x – 5h ) – ( x^2 – 5x ) = 2xh + h^2 – 5h$ .

$\dfrac{g(x + h) – g(x)}{h} = \dfrac{2xh + h^2 – 5h}{h} = 2x + h – 5$ .

Practice 3: (a) Friend (b) Friend (c) At $t = 40$ . Before that your friend is walking faster and increasing the distance between you. Then you start to walk faster than your friend and start to catch up. (d) Friend. You are walking faster than your friend at $t = 50$ , but you still have not caught up.

Practice 4: (a) The x–coordinate is increasing. (b) The x–increment $∆x$ is decreasing. (c) The slope of the line through $P$ and $Q$ is decreasing.

Practice 5: See Fig. 31.