Lesson 6 Different Combinations Practice Understanding

Ready

One hundred forty-four college freshmen were given a math placement exam with $100$ possible points. The results show that $56$ different scores were earned, ranging from $24$ to $96$ . The scores were grouped in intervals as shown in the table:

Score interval	Midpoint of interval	Frequency of interval
$92.5 – 97.5$	$95$	$2$
$87.5 – 92.5$	$90$	$4$
$82.5 – 87.5$	$85$	$10$
$77.5 – 82.5$	$80$	$13$
$72.5 – 77.5$	$75$	$21$
$67.5 – 72.5$	$70$	$26$
$62.5 – 65.5$	$65$	$18$
$57.5 – 62.5$	$60$	$15$
$52.5 – 57.5$	$55$	$12$
$47.5 – 52.4$	$50$	$8$
$42.5 – 47.5$	$45$	$3$
$37.5 – 42.5$	$40$	$3$
$32.5 – 37.5$	$35$	$4$
$27.5 – 32.5$	$30$	$4$
$22.5 – 27.5$	$25$	$1$

1.

Make a histogram of the grouped data in the chart. (Note: The midpoint of each cell is given in the horizontal axis. The sides of the cells will match the score interval. Frequency is the vertical height.)
Locate the midpoint at the top of each cell in your histogram and connect each consecutive midpoint with straight line segments. The resulting figure is called a frequency polygon. If you smooth the line segments out into a smooth curve, you will create a frequency curve. Make a frequency curve on your histogram. It should look something like the figure.

Set

Find functions $f$ and $g$ so that $f (g (x)) = H$ .

2.

$H (x) = \sqrt{x^{2} + 5 x - 4}$

3.

$H (x) = {(3 - \frac{1}{x})}^{2}$

4.

$H (x) = {(3 x - 7)}^{4}$

5.

$H (x) = | 5 x^{2} - 78 |$

6.

$H (x) = \frac{2}{3 - x^{5}}$

7.

$H (θ) = {(\tan θ)}^{2}$

8.

$H (θ) = \tan (θ^{2})$

9.

$H (x) = \sqrt{\frac{1}{6 x}}$

10.

$H (x) = 9 (4 x - 8) + 1$

Go

Use the laws of logarithms to write each logarithmic expression as the sum or difference of logarithms. Express powers as factors.

11.

$\log_{5} (3 (x + 16))$

12.

$\log_{3} (x^{4} (x^{2} - 7))$

13.

$\log_{2} (\frac{x^{3}}{x - 3})$

Use the laws of logarithms to write each logarithmic expression as a single logarithm.

14.

$\log_{9} (3 x + 2) + \log_{9} (x)$

15.

$2 \log_{7} (5 x) - \log_{7} (x + 2) + \log_{7} (x - 1)$

16.

$\ln (\frac{x}{x - 1}) + \ln (\frac{x + 1}{x})$

In problems 12–17, several of the functions studied in NC Math 1 through NC Math 3 are identified.

For each function, identify the:

Domain
Range
Intervals of increase and decrease
Whether it’s even, odd or neither
Finally, describe the graph and any interesting feature of the graph.

17.

constant function: $f (x) = b$ , where $b$ is a real number.

18.

direct variation function: $f (x) = k x$ , where $k \neq 0$

19.

inverse variation function: $f (x) = \frac{k}{x}$ , where $k \neq 0, x \neq 0$

20.

square root function: $f (x) = \sqrt{x}$

21.

absolute value function: $f (x) = | x |$

22.

cubic function: $f (x) = x^{3}$