Lesson 4 Composing and Decomposing Develop Understanding

Ready

A school building is kept warm only during school hours, in order to save money. The figure shows a graph of the temperature, $G$ , in $° F$ as a function of time, $t$ , in hours after midnight. At midnight $(t = 0)$ , the building’s temperature is $50 ° F$ ‌. This temperature remains the same until $4$ a.m. Then the heater begins to warm the building so that by $8$ a.m., the temperature is $70 ° F$ ‌. That temperature is maintained until $4$ p.m., when the building begins to cool. By $8$ p.m., the temperature has returned to $50 ° F$ ‌ and will remain at that temperature until $4$ a.m.

1.

In January, many students are sick with the flu. The custodian decides to keep the building $5 ° F$ warmer. Sketch the graph of the new schedule on the figure.

2.

If $G = f (t)$ is the function that describes the original temperature setting, what would be the function for the January setting?

3.

In the spring, the drill team begins early morning practice. The custodian then changes the original setting to start $2$ hours earlier. The building now begins to warm at $2$ a.m. instead of $4$ a.m. and reaches $70 ° F$ at $6$ a.m. It begins cooling off at $2$ p.m. instead of $4$ p.m. and returns to $50 ° F$ at $6$ p.m. instead of $8$ p.m. Sketch the graph of the new schedule on the figure.

4.

If $G = f (t)$ is the function that describes the original temperature setting, what would be the function for the spring setting?

Set

In our work, we use the notation $f (g (x))$ for function composition.

5.

Let $f (x) = 2 x^{2} - 4$ and $g (x) = 5 x$ .

a.

$f (g (1)) =$

b.

$g (f (1)) =$

6.

Let $f (x) = \frac{8}{x - 3}$ and $g (x) = \frac{15}{x + 1}$ . Find each.

a.

$f (g (x)) =$

b.

$g (f (x)) =$

7.

Use your answers for parts a and b in problem 6 to calculate the two problems given.

a.

$f (g (- 1)) =$

b.

$g (f (3)) =$

8.

Now use $f (x) = \frac{8}{x - 3}$ and $g (x) = \frac{15}{x + 1}$ to calculate $f (g (- 1))$ and $g (f (3))$ .

a.

Describe the problem that you encountered when calculating $f (x)$ and $g (x)$ separately.

b.

Do you think that the answer you derived in #7 is valid based on what happened in #8? Explain your thinking.

9.

Describe the domains of $f (g (x))$ and $g (f (x))$ .

a.

Domain of $f (g (x))$

b.

Domain of $g (f (x))$

10.

What makes the domain for each composition different?

Go

Change each logarithmic expression to an equivalent expression involving an exponent.

11.

$\log_{6} 216 = 3$

12.

$\log_{7} (\frac{1}{49}) = - 2$

13.

$\log_{3} (\frac{1}{81}) = - 4$

14.

$\log_{π} x = 2.7$

15.

$\ln x = 9$

16.

$\ln 9 = x$