1.5 Exponential and logarithmic functions  (Page 10/17)

h

g

$h\circ f$

$g\circ f$

$f(x)=2x^2+9x-5$

$f(x)=x^3+2x^2-2x$

$\frac{{\text{tan}}^{2}x}{{\text{sec}}^{2}x}+{\text{cos}}^{2}x$

$\text{cos}\left(2x\right)={\text{sin}}^{2}x$

$6{\text{cos}}^{2}x-3=0$

${\text{sec}}^{2}x-2\text{sec}x+1=0$

$5^x=16$

${\text{log}}_2(x+4)=3$

$f(x)=x^2+2x+1$

$f(x)=\frac{1}{x}$

$f(x)=\sqrt{9-x}$

$f(x)=x^2+3x+4$

A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

a. Find the equation $C=f(x)$ that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each.

a. Find the inverse function $x=f^{\mathrm{-1}}(C)$ and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has $8000 to spend.

The population can be modeled by $P(t)=82.5-67.5\text{cos}\left[(\pi \text{/}6)t\right],$ where $t$ is time in months $(t=0$ represents January 1) and $P$ is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as $P(t)=82.5-67.5\text{cos}\left[(\pi \text{/}6)t\right]+t,$ where $t$ is time in months ( $t=0$ represents January 1) and $P$ is population (in thousands). When is the first time the population reaches 200,000?

If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?